math.SP — Pith

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math.AP 2026-07-02

Any domain's Dirichlet gaps sum to at least a Bessel constant

by Yanyang Li, Quanyu Tang +1 more

The Ashbaugh--Benguria reciprocal-gap conjecture for Dirichlet eigenvalues

The inequality holds with equality exactly for balls, settling the reciprocal-gap conjecture in all dimensions.

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We prove the Ashbaugh--Benguria reciprocal-gap conjecture for the Dirichlet Laplacian in every dimension $N\ge2$. Specifically, if $\Omega\subset\mathbb R^N$ is a bounded domain and $$ 0<\lambda_1(\Omega)<\lambda_2(\Omega)\le\lambda_3(\Omega)\le\cdots $$ are its Dirichlet eigenvalues, then $$ \sum_{i=1}^{N} \frac{\lambda_1(\Omega)} {\lambda_{i+1}(\Omega)-\lambda_1(\Omega)} \ge \frac{N}{j_{N/2,1}^2/j_{N/2-1,1}^2-1}, $$ where $j_{\mu,1}$ denotes the first positive zero of the Bessel function $J_\mu$ of the first kind of order $\mu$. We also characterize the equality case: equality holds precisely when $\Omega$ agrees with a Euclidean ball up to a set of Sobolev $H^1$-capacity zero. In particular, among bounded Lipschitz domains, equality holds if and only if $\Omega$ is a Euclidean ball.

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math.SP 2026-07-01

First-variation formula for eigenvalues holds at essential spectrum edge

by Denis Vinokurov

Eigenvalue optimization via a first-variation formula

Clarke subdifferential supplies a tool that characterizes all optimal weights for weighted Laplace and Steklov problems.

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We compute the Clarke subdifferential of the $k$th eigenvalue functional on the space of self-adjoint operators, obtaining a first-variation formula that remains valid even when the eigenvalue lies at the edge of the essential spectrum. This formula provides an effective tool for describing the structure of critical points in eigenvalue optimization problems and can also yield simple proofs of the existence of optimizers. We illustrate these advantages through applications to the optimization of weighted Laplace and Steklov eigenvalues. In particular, we characterize all optimal weights, thereby answering some open questions posed by Kokarev, and give a short proof that such weights exist.

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math.AP 2026-07-01

Neumann Ornstein-Uhlenbeck semigroup on trees is Markovian with Gaussian invariant

by Sahiba Arora, Marjeta Kramar Fijavž +2 more

Ornstein--Uhlenbeck semigroup on rooted trees

Form methods on rooted metric trees yield a Markovian Neumann realization whose unique invariant is the Gaussian measure, plus spectral redu

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We study Ornstein--Uhlenbeck operators on rooted metric trees equipped with a Gaussian-type measure. Using form methods, we construct Dirichlet and Neumann realisations corresponding, respectively, to killing and reflection at the root. The associated semigroups are symmetric, analytic and positivity preserving; the Dirichlet semigroup is sub-Markovian, while the Neumann semigroup is Markovian and admits the Gaussian measure as its unique invariant measure up to scalar multiples. We prove compactness of the resolvent and derive linear eigenvalue asymptotics. For regular rooted trees, we adapt the Naimark--Solomyak decomposition to the Gaussian weighted setting, reducing the operators to one-dimensional half-line problems and obtaining refined spectral localisation and lower bounds.

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math.SP 2026-07-01

S-spectrum yields bisectorial estimates for Dirac S-resolvent

by Ivan Beschastnyi, Fabrizio Colombo +2 more

The S-resolvent estimates for the Spinor Dirac operator on manifolds with boundary conditions

It covers non-self-adjoint cases on manifolds by including right eigenvalues where earlier notions fall short.

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The aim of this paper is to show that the spectral theory based on the S-spectrum is particularly well suited for the Dirac operator on manifolds, even in cases where the operator is not self adjoint. Traditionally, for non-self adjoint operators in the Clifford setting, the literature has often referred to the right spectrum. However, a more comprehensive approach is provided by the theory of the $S$-spectrum, which is the appropriate notion for general operators on Clifford modules. In this work, we show that this theory is particularly well suited for bisectorial Clifford operators. By using the $S$-spectrum, which naturally contains the right eigenvalues, we prove bisectorial estimates for the $S$-resolvent associated with the spinor Dirac operator under various boundary conditions.

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math.AP 2026-07-01

Pleijel bound holds for p-Laplacian nodal domains

by Vladimir Bobkov

On Pleijel-type nodal domain bounds for the p-Laplacian

Cogenus eigenvalues satisfy an asymptotic upper limit on the number of domains as their index grows large

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We provide an upper estimate \`a la Pleijel on the asymptotic number of nodal domains for eigenfunctions corresponding to the cogenus eigenvalues $\{\lambda_k(p;\Omega)\}$ of the $p$-Laplacian in a bounded domain $\Omega$, and identify regimes when the number of nodal domains of the $k$-th eigenfunction is less than $k$ as $k \to +\infty$. As auxiliary results, which also have independent interest, we provide a useful characterization of the cogenus eigenvalues implying their continuity with respect to $p$, justify the Weyl law, and prove the inequality $\lambda_2(p;B) \leq \dots \leq \lambda_{N+1}(p;B) \leq \lambda_\ominus(p)$ in an $N$-dimensional ball $B$, where $\lambda_\ominus(p)$ is an eigenvalue whose eigenfunction has a central section of $B$ as its nodal set.

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math.SP 2026-07-01

Zero LE yields capacity convergence along rational frequency sequences

by Burak Hatinoğlu, Svetlana Jitomirskaya

Capacity and measure approximations for Schr\"{o}dinger operators

For continuous potentials the logarithmic capacity of phase-union spectra at rationals approaches that of the irrational quasiperiodic spect

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We prove that logarithmic capacity convergence for phase-union spectra of quasi-periodic Schr\"{o}dinger operators in the zero Lyapunov exponent regime is robust, requiring only continuity of the potential. Let $S^+(p/q)$ denote the union, over the phase, of the spectra at rational frequency $p/q$. We show that if the Lyapunov exponent vanishes on the spectrum $\Sigma(\alpha)$ at an irrational frequency $\alpha$, then for every sequence $p_n/q_n\to\alpha$, the logarithmic capacities $Cap(S^+(p_n/q_n))\longrightarrow Cap(\Sigma(\alpha)).$ We also prove convergence of the corresponding harmonic measures. As a consequence, the equilibrium measures of $S^+(p_n/q_n)$ converge in the weak$^*$ topology to the density of states measure of the quasi-periodic Schr\"odinger operator. We extend these results to multi-frequency Schr\"odinger operators and prove analogous convergence theorems, for logarithmic capacity, harmonic measure, and equilibrium measure, for ergodic Schr\"odinger operators in a general setting where the almost sure spectrum is approximated in the Hausdorff metric by union spectra of periodic operators. This abstract formulation applies, in particular, to uniformly almost periodic potentials along sequences of almost periods. We also provide counterexamples when the limiting frequency is rational.

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math-ph 2026-06-30

Sextic oscillator degeneracies match generalized Hermite zeros

by Davide Guzzetti, Dmitrii Rachenkov

Generalized Hermite Polynomials and Spectral Degeneracies of a Singular Sextic Oscillator

Discriminant factorization places level crossings at the poles of rational Painlevé IV solutions

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We study a quasi-exactly solvable singular sextic oscillator and its algebraic spectrum. For a distinguished range of parameters, we prove that the discriminant of the characteristic polynomial of the matrix determining the algebraic spectrum admits a natural factorization into three factors. One of these factors is the square of a generalized Hermite polynomial $H_{m,n}$, whose zeros are poles of a rational solution of the fourth Painlev\'e equation. Hence, the spectral degeneracies (level crossing points) corresponding to a component of the discriminant locus are in exact correspondence with the zeros of generalized Hermite polynomials, providing an exact Painlev\'e IV analogue of the Shapiro--Tater asymptotic correspondence originally conjectured for the quartic oscillator and Painlev\'e II. We also characterize the values of the parameters for which the sextic oscillator admits simultaneously two quasi-polynomial eigenfunctions with opposite exponential behaviour at infinity, and show that this phenomenon is also governed by generalized Hermite polynomials. Our result also yields a new determinantal representation of $H_{m,n}$ as the resultant of the characteristic polynomials of two complementary blocks of the matrix determining the algebraic spectrum.

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math-ph 2026-06-30

Random boson model switches from type-I to type-III BEC with interaction strength

by C. Boccato, J. Kerner +2 more

Existence and absence of Bose-Einstein condensation in the interacting random Kac-Luttinger model

Weak coupling condenses particles into the ground state; strong coupling spreads them across many states with none dominant.

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In this paper, we study interacting bosons at zero temperature in a random and higher-dimensional continuum model introduced by Kac and Luttinger. For weak interactions we prove that there is condensation in the lowest eigenstate of the one-particle Hamiltonian (type-I BEC). For strong interactions, however, we show that condensation in a localized state cannot occur. We also prove generalized condensation, where a family of eigenstates of the one-particle Hamiltonian is macroscopically occupied as a whole. Combining these results yields a scenario where there is generalized condensation into a family of eigenstates of the one-particle Hamiltonian, but none of them is macroscopically occupied itself (type-III BEC). This proves a transition in the type of condensation. To the best of our knowledge, this is the first rigorous result in this direction for a random continuum model in higher dimensions.

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math.SP 2026-06-30

Neumann-Dirichlet eigenvalue shift grows exponentially with dimension

by N. Filonov

Inequalities between Dirichlet and Neumann eigenvalues in large dimensions

Ψ is at least C(e/2)^d for the first eigenvalue on any bounded domain and for all eigenvalues on convex domains.

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Let $\Omega$ be a bounded domain in $R^d$. Denote by $\lambda_k$ (resp. $\mu_k$) the eigenvalues of the Laplace operator in $\Omega$ with Dirichlet (resp. Neumann) boundary conditions. Denote by $\Psi = \Psi (d,k,\Omega)$ the shift of indices in the inequality $\mu_{k+\Psi} \le \lambda_k$. We are interested to describe the behaviour of $\Psi$ for large $d$. We prove that a) $\Psi (d,1,\Omega) \ge C (e/2)^d$ for all domains $\Omega$; and b) $\Psi (d,k,\Omega) \ge C (e/2)^d$ for all $k$ and all convex domains $\Omega$.

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math.AP 2026-06-29

Local DN equality extends to full boundary if metrics agree nearby

by Thierry Daudé, Alberto Enciso +3 more

A Local--to--Global Propagation Principle for Dirichlet--to--Neumann Maps

For Riemannian metrics coinciding in a collar, matching local Dirichlet-to-Neumann data on an open set implies global agreement on the compo

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We establish three local-to-global propagation results for Dirichlet--to--Neumann maps. First, in a general geometric setting, we show that if two smooth Riemannian metrics coincide in a collar neighborhood of a connected boundary component $\Gamma$, then equality of the corresponding local Dirichlet--to--Neumann maps on a nonempty open subset of $\Gamma$ propagates to equality of the associated global Dirichlet--to--Neumann maps on all of $\Gamma$. The proof combines unique continuation and self-adjointness arguments. Our second result replaces the geometric collar assumption by an exponential spectral assumption on the difference of the corresponding global Dirichlet--to--Neumann maps. The proof relies on the spectral unique continuation theory of Jerison--Lebeau, through the formulation of Le~Rousseau--Lebeau. Finally, we specialize to a particular class of conformally warped product metrics. In this setting, the local Borg--Marchenko theorem identifies the exponential spectral assumption with the coincidence of the metrics in a collar neighborhood of the boundary. Assuming in addition that the boundary is a compact Riemannian symmetric space, we show that this assumption can be substantially weakened by requiring only a suitable quasi--analytic boundary closeness of the conformal factors. The proof combines Weyl--Titchmarsh theory with the quasi--analytic propagation theorem of Ganguly and Thangavelu.

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math.SP 2026-06-29

Transfer operator framework yields absolute continuity of IDS

by Xianzhe Li, Zhenfu Wang +2 more

Transfer Operators, Canonical Center Dynamics, and Spectral Applications for Long-Range Operators

For quasi-periodic Schrödinger operators the center bundle reduces the spectral problem and implies Anderson localization.

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We introduce an operator-theoretic framework for long-range operators over general dynamical systems with analytic hopping and small potential. By establishing a partially hyperbolic splitting on the fibered solution bundle, we define the Canonical Center Bundle (CCB) as the center subbundle of this splitting, which is shown to be globally trivial. The center bundle admits a representation via Riesz spectral projections of the transfer operator. Furthermore, we show that, in the local regime, the center bundle arising in this framework essentially coincides, in the sense of gap convergence, with the Intrinsic Center Bundles (ICB) obtained from finite-range approximations in \cite{GJ}. The partially hyperbolic structure thereby reduces the spectral problem to the center bundle, leading to a Johnson-type characterization of the spectrum in terms of the associated center cocycle. We then apply this framework to quasi-periodic Schr\"odinger operators with analytic hopping, large analytic potentials and Diophantine frequency. In this setting, the center cocycle is analytic and satisfies a Center Thouless formula. As consequences, we establish the absolute continuity of the integrated density of states (IDS), resolving a problem of Eliasson; prove quantitative H\"older continuity of the IDS, partially answering a question of You; and obtain Anderson localization for the original Schr\"odinger operators.

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math.SP 2026-06-29

Discrete Prüfer phase yields O(h²) eigenvalue accuracy

by F. Ayça Çetinkaya, Kürşat Er +1 more

A Discrete Pr\"ufer Transformation Approach to Sturm--Liouville Difference Equations and Eigenvalue Estimation

An exact algebraic recurrence for the phase enables shooting that converges to continuous eigenvalues at second-order rate.

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In this paper, we study regular second-order Sturm--Liouville difference equations using the discrete Pr\"ufer transformation. By representing solutions in amplitude and phase coordinates, we analyze an exact algebraic phase system that guarantees unique, monotonic phase tracking and preserves classical oscillation properties. Using this theoretical foundation, we develop a Pr\"ufer-based numerical shooting method to compute eigenvalues for discrete boundary value problems. To initialize the root-finding algorithm, we apply Gershgorin's theorem to the difference operator to establish mathematically guaranteed starting search intervals. Numerical experiments on classical benchmark problems demonstrate that the proposed method effectively isolates the discrete spectrum and converges to the exact continuous eigenvalues with second-order $\mathcal{O}(h^2)$ accuracy.

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math.SP 2026-06-29

Stabilization cost rate equals Lyapunov exponent on Aubry-André chain

by Nassim Athmouni, Nejib Brahmia +1 more

The cost rate of nonlinear remote stabilization on the Aubry--Andr\'e lattice: a reflected off-spectral exponent and the sharp identity for almost every phase

The energy needed to control a distant site grows at the off-spectral cocycle rate for every phase when coupling is subcritical or critical.

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We study the exponential rate $r(\alpha,\lambda)$ of the energy $\mathcal{E}_N$ needed to steer a far site, at distance $N$, of an Aubry--Andr\'e chain $H_\lambda$ via one boundary actuator with closed-loop margin $\alpha$. An exact eigenbasis reduction writes $\mathcal{E}_N$ as a Cauchy quadratic form $\tilde b^\top C^{-1}\tilde b$ in the boundary-amplitude ratios, whose rate is the off-spectral Lyapunov exponent of the transfer cocycle at the reflected band edge $z^\star=2E_{\min}-2\alpha-E_{\max}$, giving $r(\alpha,\lambda)=\gamma_\lambda(z^\star)$. The rate lies in a bracket of width $\log_+(\lambda/2)$ whose ends coincide for $\lambda\le2$, the spectrum having logarithmic capacity $\max(1,\lambda/2)$. We prove the identity unconditionally, for every phase, on the whole metallic--critical range $0<\lambda\le2$: for $\lambda<2$ through subcritical almost reducibility as the sole external input, and at $\lambda=2$ because the Green's function there equals the Lyapunov exponent. For $\lambda>2$ the upper bound is unconditional, and the lower bound takes localization as its only external input: an inverse-free cocycle form makes $\mathcal{E}_N$ a cancellation-free positive sum, and a Christoffel--Darboux identity collapses its coefficients to $|c_k|=Q(\delta_k)(\hat\psi^{(k)}_N)^2$, where band-edge near-degeneracies cancel. With a three-distance lemma this yields $r=\gamma_\lambda(z^\star)$ at every Diophantine frequency and almost every phase, with gap $O(N^{-2/(2+\tau)})$ for type $\tau$ ($N^{-2/3}$ at bounded type), unconditionally for $\lambda\ge\lambda_1$ and under a polynomial-prefactor localization hypothesis for $2<\lambda<\lambda_1$. The relative gap $1-r/\gamma_\lambda(z^\star)$ vanishes at both ends of the localized phase, with $g_{\mathbb{C}\setminus\Sigma_\lambda}(z^\star)\to\operatorname{arccosh}3$ as $\lambda\to\infty$.

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math.FA 2026-06-29

Kato-Rosenblum theorem extends to unbounded n-tuples

by Rhishab Bhutani, Dan Virgil Voiculescu

A Sharp Kato-Rosenblum Type Theorem for Unbounded n-Tuples

Commuting self-adjoint operators whose difference lies in the Lorentz (n,1) ideal share the same absolutely continuous spectrum when n is at

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We prove a generalization for commuting $n$-tuples of unbounded self-adjoint operators and the Lorentz $(n,1)$ ideal,$n \ge 3$, of the Kato-Rosenblum theorem. The result is derived from earlier work for bounded operators [8]. Also, a very weak result for $n=2$ unbounded operators and other additional results are obtained.

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math.SP 2026-06-29

Perturbations yield Breit-Wigner resonances near embedded eigenvalues

by Hemant Bansal, Alok Maharana +1 more

Multi-parameter Perturbations of the Laplacian and Resonance Near a Simple Embedded Eigenvalue

Asymptotics for spectral density, cross-section and time delay hold as the multi-parameter vector approaches the reference operator.

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This paper continues the study of resonance phenomena initiated in [3] for rank-one perturbations. We consider finite-rank multi-parameter perturbations $H_\alpha$ of the Laplacian on $L^2(\mathbb{R}^3)$ and establish Breit--Wigner-type asymptotics for the spectral density of $H_\alpha$ along the resonance $\lambda(\alpha)$ near a simple embedded eigenvalue $\lambda_0$ of $H_a$ as $\alpha\to a$. We also obtain similar asymptotic behaviour for the scattering cross-section and the average time delay.

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math.NT 2026-06-29

Automorphic forms split into cusp forms plus Eisenstein coefficients

by Devadatta G. Hegde

On Franke's theorem in the simplest case

Direct proof for level one on the half-plane uses only growth conditions and Green's identity, bypassing Langlands spectrum construction.

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For level one spherical automorphic forms on the upper half-plane, we prove directly that every automorphic form is a sum of a cusp form and a linear combination of Laurent coefficients of the standard Eisenstein series. This is the simplest instance of Franke's general theorem, which asserts that automorphic forms on a reductive group are spanned by Laurent coefficients of Eisenstein series induced from cuspidal automorphic forms on Levi subgroups. Unlike Franke's general argument, ours does not invoke Langlands' construction of the discrete automorphic spectrum from cuspidal Eisenstein series. It rests instead on basic analytic properties of automorphic forms and Green's identity.

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math-ph 2026-06-26

Z2 edge index at curved boundary equals bulk difference times intersection mod 2

by Alexis Drouot, Jacob Shapiro +1 more

Geometric bulk-edge correspondence for mathbb{Z}₂-topological insulators

Gives explicit rule for protected states along arbitrary smooth interfaces between time-reversal invariant insulators.

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Fermionic time-reversal-invariant insulators in two dimensions--class AII in the Kitaev table--come in two topological phases. These phases are characterized by a $\mathbb{Z}_2$-valued invariant, the Fu-Kane-Mele index. We prove a geometric bulk-edge correspondence for curved interfaces: if two such insulators occupy complementary regions separated by a curved boundary, then the $\mathbb{Z}_2$ edge index of the interface system is the product, modulo two, of the difference of the two bulk $\mathbb{Z}_2$ indices and a geometric intersection number associated with the boundary and the measurement region. The argument is a $\mathbb{Z}_2$ analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}.

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math.SP 2026-06-26

Wave kernels on trees written as finite I-Bessel sums

by Amar Bašić, Lejla Smajlović +1 more

Discrete Space-Time Wave Kernels on Regular Trees

Nonnegativity on the generalized Laplacian yields convolution solutions and a J-Bessel form at the spectral bottom.

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We study the forward discrete space-time wave equation on the homogeneous $(q+1)$-regular tree $T_{q+1}$ associated with a two-parameter generalized Laplacian. Under the natural nonnegativity assumption on this operator, we derive explicit formulas for the two fundamental wave kernels. The formulas are given in terms of discrete $I$-Bessel functions and yield convolution representations for solutions with general initial conditions. In the boundary case corresponding to the bottom of the spectrum, we obtain another explicit representation of the wave kernel in terms of discrete $J$-Bessel functions. This representation leads to a discrete analogue of the classical $I\!\leftrightarrow\!J$ relation. We also perform both analytic and numerical studies of the asymptotic behavior of the wave kernels, including large radial distance, large time, and large degree of the tree. An important feature of our analysis is that the wave kernels are expressed as finite sums; hence, the propagation formulas remain finite for every discrete time.

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math.CO 2026-06-26

Wave kernels on graphs equal Bessel functions of walk counts

by Amar Bašić, Lejla Smajlović +1 more

Discrete Space-Time Wave Kernels and Trace Identities on Regular Graphs

This yields trace identities and closed forms for twisted trigonometric and Chebyshev sums on regular graphs.

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We study the discrete space-time wave equation on a $(q+1)$-regular graph $X$ associated with the affine Laplace-type operator. For the forward time-difference scheme we derive explicit formulas for the two fundamental solutions (wave kernels) in terms of discrete modified Bessel functions and the non-backtracking walk counts on $X$ thus providing a direct and explicit link between wave propagation and combinatorial graph data. Utilizing uniqueness property of the wave kernel, we prove a new trace-type formula associated to the affine Laplace-type operator on $X$ and apply it to deduce many combinatorial identities. For example, we derive a closed-form expression for evaluation of some trigonometric sums twisted by an additive character as well as evaluations of finite sums of Chebyshev polynomials twisted by binomial coefficients.

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math.FA 2026-06-26

Duality preserves Morse critical groups for ratio convex functions

by Dong Zhang

Hidden critical and Morse equivalence behind duality: Theory and Applications

Polarity dual keeps sublevel homotopy, critical groups, and handle decompositions unchanged for RC functions and yields a decomposition-free

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The aim of this paper is to establish critical duality theory for ratios of nonnegative homogeneous convex functions (shorten for RC functions) and differences of convex functions (abbreviated as DC functions) on Banach spaces. Specifically, we establish a series of duality results on critical point theory and Morse theory for RC functions, including the homotopy type of sublevel sets, the Morse critical points and their Rothe critical groups, Lagrange critical points and their multiplicities, Lusternik-Schnirelman min-max critical values, Poincare polynomials, as well as the structure of handlebody decompositions, all of which are proved to be preserved under polarity dual. Moreover, we obtain the first critical duality theory of DC functions which does not depend on the DC decomposition. This answers a question left open from the work of Toland on DC functions and the work of Le-Pham on DC programming. We apply these results to provide a reformulation of the graph Cheeger constant using zonotopes; we introduce the contact data which serves as a geometric characterization of Lagrange criticality; and we show that the eigenproblems for 1-Laplacian and $\infty$-Laplacian on hypergraphs are equivalent to the contact problems of zonotopes, which indeed establishes a new characterization of zonotopes. We also prove a duality equivalence for certain nonlinear eigenvalue problems and bifurcation problems. Our study here reveals an intricate interaction of critical point theory with other fields such as convex analysis, combinatorial geometry, and nonlinear eigenproblems on graphs.

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math.PR 2026-06-26

KPZ exponent controls second term of LQG heat trace

by Nathanaël Berestycki, Jakob Klein

Spectral expansion of LQG heat trace and KPZ scaling

Expected value of the domain integral of the on-diagonal heat kernel obeys the predicted scaling as time vanishes.

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Let $h$ be a whole plane Gaussian free field, and let $\Omega$ be a bounded domain in two dimensions. We study the asymptotics as $t\to 0$ of the Liouville quantum gravity (LQG) heat trace, defined as the integral over $\Omega$ of the on-diagonal LQG heat kernel. Our main result is to show that the second term in the spectral expansion as $t\to 0$ of the expected heat trace is governed by a nontrivial exponent, given by the KPZ (Knizhnik--Polyakov--Zamolodchikov) relation. A similar but stronger (almost sure) result applies to the related notion of heat content. Along the way we obtain various results on the short-term behaviour of the heat kernel, notably solving a conjecture of \cite{BW} concerning its annealed asymptotics, and showing the finiteness of all moments of the properly rescaled heat kernel.

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math.AP 2026-06-26

Resolvent estimates for inverse-square potentials sharpen with large coupling

by Piero D'Ancona, Jérémy Faupin +1 more

Quantitative uniform resolvent estimates

A matrix representation of weighted resolvent boundaries yields the estimates via partial wave decomposition.

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We derive quantitative uniform resolvent estimates for Schr\"odinger operators on the half-line with inverse-square potentials, which provide a sharp behaviour in the limit of large coupling. Our approach is based on a matrix representation of the boundary value of a weighted resolvent. The partial wave decomposition then turns these one-dimensional channel estimates into explicit weighted resolvent estimates for the Laplacian, its inverse-square potential perturbations and for the magnetic Laplacian with an Aharonov--Bohm potential. We also obtain exact Simon-type identities for the imaginary parts of the weighted resolvents of these operators.

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math.SP 2026-06-26

One scattering entry determines multilayer domain and metric

by Michel Cristofol (AMU, I2M) +4 more

Inverse scattering in an asymptotically flat multilayer domain

The full geometry and coefficients are recovered from the diagonal component of the S-matrix on a single flat slab at all energies.

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We consider a scattering problem for a wave equation $\partial_t^2 u = \frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij}\partial_j)u$ in a multilayer domain $\Omega \subset {\bf R}^{n+1}_x = {\bf R}^n_y \times {\bf R}^1_{x^{n+1}}$ of the form $\Omega = \mathcal K \cup \Omega_1 \cup \cdots \cup \Omega_N$, where $\mathcal K$ is a bounded open set and $\Omega_k$ is asymptotically equal to a slab domain ${\bf R}^n \times (c_k,c_k + d_k)$ as $|y| \to \infty$. Assuming that $\partial_x^{\alpha}\big(g_{ij}(x) - \delta_{ij}\big) = O(|x|^{-|\alpha| - \delta_0}), \ \delta_0 > 1, \forall \alpha$, we show that $\Omega$ and $g^{ij}$ are determined by one diagonal component $S_{11}(\lambda)$, for all energies, of the S-matrix associated with the slab $\Omega_1$, provided $\Omega_1$ is flat: $\Omega_1 \cap \{|y| > R\} = \{|y| > R\} \times (c_1, c_1+d_1)$ for some constants $c_1, d_1, R > 0$, and the metric is Euclidean on $\Omega_1\cap \{|y| > R\}$.

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math.AP 2026-06-26

Multilinear oscillatory bounds reach sharp endpoints and drop logs from 3D spectral cluste

by Shengwen Gan, Cheng Zhang +1 more

Sharp endpoint multilinear estimates for oscillatory integrals and spectral clusters

The result gives log-free bilinear cluster estimates on every closed three-dimensional manifold and sharp multilinear versions for all k and

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We prove sharp $k$-linear $L^p$ estimates for Carleson--Sj\"olin oscillatory integral operators with arbitrary separated frequency scales for all $k\ge 2$ and $1\le p\le \infty$. The estimates are sharp, including the endpoint logarithmic behavior for general Carleson--Sj\"olin phases. Moreover, we obtain log-free endpoint bilinear spectral cluster estimates on every closed three-dimensional Riemannian manifold, resolving a problem of Burq--G\'erard--Tzvetkov. As a consequence, we establish sharp $k$-linear $L^p$ spectral cluster estimates for all $k\ge 2$ and $1\le p\le \infty$.

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math.DG 2026-06-24

CR Paneitz operator has infinitely many negative eigenvalues on non-embeddable tori

by Pak Tung Ho, Yuya Takeuchi

Non-embeddable torus and CR Paneitz operator

The spectral result holds under mild assumptions on three-dimensional tori that cannot be embedded.

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The CR Paneitz operator is closely related to several important problems in CR geometry. In this paper, we study the CR Paneitz operator on non-embeddable three-dimensional tori. Under mild assumptions, we show that it possesses infinitely many negative eigenvalues. We also provide concrete examples satisfying the assumptions.

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math.DG 2026-06-24

Off-diagonal expansion holds for indicator-symbol Toeplitz kernels

by Razvan Apredoaei

Asymptotics for Toeplitz operators with symbol an indicator function

This allows extending trace asymptotics of polynomials and Weyl laws to non-compact symplectic manifolds of bounded geometry.

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We prove an off-diagonal expansion of the kernel of the Toeplitz operator whose symbol is the indicator function of a compact domain with smooth boundary in a complete symplectic manifold of bounded geometry. Using our approach, we extend two results to the non-compact setting: the first concerns the asymptotics of the trace of polynomials in this operator, and the second establishes a Weyl law for this Toeplitz operator.

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math.SP 2026-06-24

Ball maximizes Robin eigenvalue for small negative coupling

by Nunzia Gavitone, David Krejcirik +1 more

A reverse Faber--Krahn inequality for the Robin Laplacian with negative boundary parameter: small coupling in all dimensions

Among convex domains of fixed volume in any dimension, the ball has the largest first eigenvalue when the boundary parameter is close to zer

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We establish Bareket's conjecture from 1977 for convex domains in all dimensions in the regime of weak boundary coupling. In other words, we consider the Laplace operator, subject to negative boundary conditions, and show that the ball maximises the first eigenvalue among all bounded convex domains of fixed volume, provided that the boundary parameter is sufficiently close to zero. The smallness depends on the volume and dimension only. The proof relies on a comparison with spherical shells with combined Neumann--Robin boundary conditions obtained via the method of parallel coordinates, which we manage to extend to all dimensions, and on a careful analysis of the corresponding radial problem.

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math.CO 2026-06-23

Equal sides uniquely maximize spanning trees in grid products at perfect powers

by Jiechen Zhang

Extremal Spanning Trees in Product Grid Graphs

Pairwise balancing theorems strengthened by heat-trace Schur-concavity give unique maximizers for free and periodic boundaries.

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We study how fixed-volume spanning-tree extremality changes when product-grid boundary factors are free, periodic, or mixed. In two dimensions, extremality depends sharply on the boundary type. The free/free and periodic/periodic products both obey a closest-to-square principle: among fixed-area rectangles, $P_r\square P_s$ and $C_r\square C_s$ are maximized by the closest-to-square admissible side lengths. The mixed free/periodic cylinder $P_r\square C_s$ is different: closest-to-square fails, and in the divisor-rich case the optimizing cyclic circumference has scale $N^{1/3}$ when the area is $N=rs$. In arbitrary dimension we prove pairwise balancing theorems for pure free products and pure periodic products, and then strengthen them by a heat-trace Schur-concavity theorem in logarithmic side lengths. At perfect-power volume this gives the unique maximizers $P_n^{\square d}$ and, for $n\ge3$, $C_n^{\square d}$. These product-grid comparisons motivate perfect-power conjectures for connected induced lattice subgraphs and periodic analogues.

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math.NA 2026-06-23

Ten-digit certification for singular Schrödinger spectrum and non-normal resonances

by Matthew J. Colbrook

Ten Digits on a Train: AI-Assisted Verification of Two Eigenvalue Problems

Global matching with projective lines and componentwise inclusions encloses eigenvalues where standard shooting fails.

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Accurate numerical eigenvalues are often difficult to certify, especially in singular or non-normal settings. This article reports a human--AI collaboration on two such computations. For a singular self-adjoint Schr\"odinger operator, a verified zero count and Dirichlet--Neumann bracketing certify the complete negative spectrum to ten decimal places. For a delicate non-normal atom--molecule benchmark, a previously unresolved resonance pair is separated, with each member enclosed to ten digits. The second result is achieved not by increasing the precision of one-way shooting, but by reformulating the problem as a global matching system for projective solution lines. The infinite tail is encoded as uncertainty in the terminal projective data, and a componentwise, tail-robust Krawczyk--Brouwer inclusion supplies the certificate. This gives a reusable architecture for analytic boundary-value systems with ill-conditioned propagation and uncertain asymptotic data. The collaboration also exposes the strengths and limits of AI assistance. AI rapidly produced accurate candidates and plausible proof strategies, but several failed, including one apparently complete tail argument that omitted the componentwise check required by a nonuniform polydisc. Validated computation is a stringent test of AI-assisted mathematics: the output is not merely a number, but a number with a proof. These examples show why the proof object matters, and why human mathematical judgment remained decisive. More broadly, as AI makes code, exposition, and plausible numerical claims inexpensive, standards for verification, attribution, peer review, and training must adapt. The implications are unsettling; the opportunity is extraordinary.

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math.DS 2026-06-23

Lyapunov exponent bounded below by c log λ for skew-shift operators

by Chao Wang, Yuanyuan Peng +1 more

Positivity and log-H\"{o}lder Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schr\"{o}dinger Operators

Uniform positivity and log-Hölder continuity in energy hold for Diophantine frequencies and large coupling in the multi-frequency setting.

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We prove the positivity and continuity of the Lyapunov exponent for one-dimensional discrete Schr\"{o}dinger operators with multi-frequency skew-shift potentials. For the operator $H_{\lambda,\omega} = \Delta + \lambda v(T_{\omega}^n(x,y))$ on $\ell^2(\mathbb{Z})$, where $T_{\omega}$ is a skew-shift on $\mathbb{T}^{d}\times\mathbb{T}^{d}$ $(d\geq1)$ and $v$ is a non-constant real-analytic function on $\mathbb{T}^{2d}$, we establish that for Diophantine frequency vectors $\omega$ and large coupling $\lambda \gg 1$, the Lyapunov exponent satisfies $L(\lambda,E) \geq c\log\lambda > 0$ uniformly in $E$ (with $c>0$), and is log-H\"{o}lder continuous in $E$. This work extends the known results of Lyapunov exponents--previously developed for one-frequency or simpler quasi-periodic models--to the genuinely multi-frequency skew-shift setting.

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math.CA 2026-06-23

Anisotropic FUP gives essential spectral gap for 2D baker's map

by Long Jin, An Zhang +1 more

Anisotropic 2D FUP and quantum open baker's map

The discrete uncertainty principle on the Bedford-McMullen carpet extends prior 1D and isotropic results under a new line porosity condition

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We prove an essential spectral gap for 2D anisotropic quantum open baker's map. This extends the 1D results of Dyatlov--Jin 2017 and the isotropic 2D results of Cohen 2025a. The key ingredient is the anisotropic discrete fractal uncertainty principle (FUP) associated with a 2D anisotropic fractal set called the Bedford--McMullen carpet. We also study the relation between our anisotropic discrete FUP and its continuous counterpart in the spirit of Dyatlov--Jin 2018 and Cohen 2025a. In particular, we prove {continuous FUP} for 2D {anisotropic porous} sets, extending the (high-dimensional) isotropic results of Cohen 2025b. To the best of our knowledge, the anisotropic (line) porosity condition -- a variant of Cohen's line porosity and stronger than ball porosity -- appears to be new to the literature.

0

math.AP 2026-06-22

Product of first N Steklov eigenvalues bounded by ω_N/|Ω| on Lipschitz domains

by Haiqi Zhang, Quanyu Tang +1 more

A sharp fixed-volume product inequality for the first N nonzero Steklov eigenvalues

The sharp fixed-volume inequality now applies to all bounded Lipschitz domains, removing the convexity requirement from earlier work.

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We prove a sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues of bounded Lipschitz domains in $\mathbb R^N$. More precisely, if $N\ge2$ and $\Omega\subset\mathbb R^N$ is a bounded Lipschitz domain, then $$ \prod_{j=1}^N \sigma_j(\Omega)\le \frac{\omega_N}{|\Omega|}, $$ where $0=\sigma_0(\Omega)<\sigma_1(\Omega)\le\sigma_2(\Omega)\le\cdots$ are the Steklov eigenvalues of $\Omega$, and $\omega_N$ denotes the volume of the unit ball in $\mathbb R^N$. This extends the convex-domain theorem of Henrot, Philippin, and Safoui to arbitrary bounded Lipschitz domains, and in particular settles the remaining higher-dimensional case of a problem posed by Henrot.

0

math.SP 2026-06-22

White-noise Schrödinger eigenvalues expand with explicit Brownian term

by Yingdu Dong, Wenwen Jian +1 more

High-energy asymptotics for finite-interval Schr\"odinger operators with Gaussian white-noise potential

k_n equals nπ/L plus stochastic integral of order 1/n plus L^p error O(n^{-2}), implying almost-sure positivity for large n

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We study the one-dimensional Schr\"odinger operator on a fixed interval with Gaussian white-noise potential, \[ H_\omega=-\frac{\dd^2}{\dd x^2}+\rho\dot B_x(\omega), \] under Dirichlet boundary conditions. The operator is defined pathwise through the quasi-derivative realization of Sturm--Liouville operators with distributional potentials. Let $\lambda_n$ be the Dirichlet eigenvalues, $\lambda_n^+=\max\{\lambda_n,0\}$, and $k_n=\sqrt{\lambda_n^+}$. For every finite $p$, we prove the high-energy expansion \[ k_n=\frac{n\pi}{L} +\frac{\rho}{n\pi}\int_0^L \sin^2\left(\frac{n\pi s}{L}\right)\,\dd B_s +O_{L^p(\Omega)}(n^{-2}). \] Consequently, almost surely, $\lambda_n>0$ for all sufficiently large $n$ and, for every $\varepsilon>0$, \[ k_n=\frac{n\pi}{L}+O(n^{-1+\varepsilon}). \] We also obtain first-order eigenfunction asymptotics with explicit Brownian stochastic-integral corrections. In particular, for the $L^2(0,L)$-normalized Dirichlet eigenfunction $\varphi_n$, with a fixed sign convention, \[ \sup_{0\le x\le L} \left|\varphi_n(x)-\sqrt{\frac{2}{L}}\sin(k_n x)\right| =O(n^{-1+\varepsilon}) \] almost surely. The proofs use stochastic Pr\"ufer coordinates, stochastic Volterra expansions, the Burkholder--Davis--Gundy inequality, and a Borel--Cantelli argument. The estimates provide a first step toward KAM-type small-divisor analysis for Hamiltonian PDEs with white-noise spatial potentials.

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math.CA 2026-06-22

Characteristic function yields explicit biorthogonal formulas for eigenparameter-dependent

by Yagub N. Aliyev, Narmin N. Aliyeva

Minimality of the root functions of Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter

Representations convert basis and minimality criteria into statements about the characteristic function and its derivatives at eigenvalues.

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We consider a Sturm--Liouville problem in which the spectral parameter appears linearly in one of the boundary conditions. The study focuses on the root functions of the problem, including eigenfunctions and associated functions corresponding to multiple eigenvalues. By employing the characteristic function of the boundary value problem, explicit representations are obtained for the biorthogonal system and for several special associated functions that play a crucial role in the spectral analysis. These representations allow previously established criteria for the basis and minimality properties of the system of root functions to be reformulated directly in terms of the characteristic function and its derivatives at the eigenvalues. As a consequence, the investigation of particular boundary value problems becomes considerably simpler. Several illustrative examples are analyzed to demonstrate the effectiveness of the proposed approach and to show its agreement with known results in the literature.

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0

math.DG 2026-06-22

Neural network recovers Morse index 4 of critical catenoid

by Miraj Samarakkody

Physics-Informed Neural Networks for Computing the Morse Index of the Critical Catenoid

Enforcing parity and treating eigenvalues as parameters yields accuracy sufficient to confirm the known index and nullity via homotopy track

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The Morse index of a free boundary minimal surface is encoded in its Jacobi-Steklov spectrum, and we test how faithfully a physics-informed neural network (PINN) reproduces that spectrum on a problem whose answer is already known in closed form. The benchmark is the critical catenoid in the unit ball $\mathbb{B}^3$, where it is well known that the Morse index equals $4$ and the nullity equals $2$. Separating the angular variable reduces the eigenvalue problem to a family of one-dimensional Robin problems on $[-T,T]$, one for each Fourier mode. A network that enforces the parity of each mode by construction, and carries the eigenvalue as a trainable parameter, returns the three eigenvalues below the stability threshold to within $10^{-6}$ to $10^{-4}$ of their exact values, with PDE residuals of order $10^{-4}$; assembling them recovers the index $4$ and the nullity $2$. We then track the spectrum along a one-parameter homotopy joining a flat reference operator to the catenoid Jacobi operator and identify the crossings at which the index changes. Since the critical catenoid is rigid, a fact we prove, this homotopy deforms operators rather than surfaces. We close by explaining how the same pipeline, with its one-dimensional solver replaced by a two-dimensional one, is poised to address genuinely geometric families in ellipsoidal balls, where the boundary curvature is no longer constant, and the Morse index is not yet known.

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math.SP 2026-06-22

Spectrum of graph Laplacian forces potential to zero

by Matthias Hofmann, Joachim Kerner

Ambarzumian-type theorems for Hermitian matrices with applications

If eigenvalues match the unperturbed discrete Laplacian on a finite graph, any added potential must vanish.

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A foundational result in inverse spectral theory due to Ambarzumian (1929) states that the Neumann Laplacian on an interval is not isospectral to the Neumann Laplacian with an additional non-zero potential. In this note, our aim is to investigate Ambarzumian-type theorems for certain classes of Hermitian matrices, including well-known matrices such as the discrete Laplacian on finite graphs. In addition, using different methods, we establish an Ambarzumian-type theorem for matrices with vanishing diagonal, in particular, the adjacency matrix on finite graphs. In this way, we generalize existing results on Ambarzumian-type theorems to general finite discrete graphs.

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math.DG 2026-06-22

Eigenvalues converge on desingularized special Lagrangian cones

by Maxwell Stolarski, Wei-Bo Su

Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow

For small scales, finite modes match the conical spectrum while a gap holds on the orthogonal complement and the scaling mode is identified.

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Let $\mathcal C$ be a $G$-invariant special Lagrangian cone admitting a scaled family of $G$-invariant special Lagrangian desingularizations $a \overline L$ which converge to $\mathcal C$ as $a\searrow 0$. We study the linearized self-shrinker operator on $a\overline L$ in a Gaussian weighted $L^2$ space of $G$-equivariant functions. For $0<a\ll1$, we construct any prescribed finite number of eigenfunctions whose eigenvalues converge to those of the limiting conical operator, and we prove a spectral gap estimate on the orthogonal complement of these modes. We also identify the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization. This spectral basis provides the analytic foundation for the construction of Type II blow-up solutions of Lagrangian mean curvature flow in the companion paper.

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math.SP 2026-06-22

Equilateral torus uniquely maximizes zeta determinant in all dimensions

by Fabio Francesconi, Julie Rowlett

Global Extrema of the Zeta Regularized Determinant on Orthogonal Flat Tori

Among unit-volume orthogonal flat tori the equilateral one is the unique maximizer for every n >= 2, and the value drops to zero at large n.

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The search for extremal geometries is a central theme in several areas of mathematics. Here, we address the following question: among all n-dimensional orthogonal tori of unit volume, which one maximizes the zeta regularized determinant of the Laplacian? We prove that the equilateral torus is the unique maximizer in each dimension n, for all n greater than or equal to 2, validating Sarnak's conjecture in this context. We also investigate the analogous question for the Laplacian on Euclidean boxes with the Neumann and Dirichlet boundary conditions. For orthogonal flat tori of unit volume and dimension n, we show further that the determinant is strictly decreasing with the dimension and tends to zero as the dimension n tends to infinity.

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math.SP 2026-06-22

Eigenvalue asymptotics for delay operators uniform in delay size

by Dmitry M. Polyakov

Spectral asymptotics for two-term even-order differential operators with homogeneous delay

A single expansion works for all indices and both Dirichlet and Neumann conditions on the unit interval.

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We consider two-term even-order operators on the unit interval with homogeneous delay and with Dirichlet and Neumann boundary conditions. The main result provides to the eigenvalue asymptotics of this operator with respect to all indices enumerating the eigenvalues. This asymptotic formula is uniform in the parameter of the homogeneous delay. We also discuss the nontrivial high-frequency phenomenon demonstrated by the uniform spectral asymptotics.

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math.PR 2026-06-22

Tridiagonal stochastic matrices have at most floor(n/2) negative eigenvalues

by Bassam Mourad, Issam Kaddoura +1 more

Negative index, matchings, and nonnegative eigenvalues of tridiagonal stochastic matrices

The bound is attained in every dimension and forces the second eigenvalue to be nonnegative for n at least 3

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We study negative eigenvalues of $n\times n$ stochastic matrices whose off-diagonal support is constrained by a sparse graph. The main tool is a matching-based inertia principle: if $G$ is bipartite with matching number $\mu(G)$, $S$ is a real symmetric matrix supported on $G$ with nonnegative diagonal entries and whose negative index (i.e. number of negative eigenvalues counted with their multiplicities) is denoted by $\nu_{-}(S) $, then \[ \nu_{-}(S)\leq \mu(G). \] In particular, every $n\times n$ nonnegative tridiagonal stochastic matrix $P$ satisfies $ \nu_{-}(P)\leq \left\lfloor \frac{n}{2}\right\rfloor. $ Consequently, after ordering the eigenvalues of $P$ in the decreasing order, we have $ \lambda_{\lceil n/2\rceil}(P)\geq0, \ \text{and hence} \ \lambda_2(P)\geq0, \mbox{ for } n\geq3. $ This gives an all-dimensional strengthening of the previously known $4\times4$ tridiagonal stochastic result. Next, we show that this tridiagonal bound is sharp in every dimension in both reducible and irreducible cases. Finally, we explore some possible extension and raise some open questions.

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math.SP 2026-06-19

Long-range Dirac-Stark operators keep eigenvalues near Stark ladders

by Moacir Aloisio, César de Oliveira +1 more

Localization and eigenvalue asymptotics for long-range discrete Dirac operators with Stark potential

Eigenfunctions obey power-law decay, so the spinorial evolution has finite position moments.

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We study long-range discrete Dirac operators with Stark potential, extending the theory of Stark localization from scalar lattice models to systems with internal spinorial structure. We initially investigate the local setting, where two distinct localization mechanisms arise. The standard local Dirac-Stark operator yields two Stark-type spectral ladders and exponentially localized spinorial eigenfunctions. Conversely, a related pure-shift local model exhibits an invariant block structure that leads to explicitly computable eigenvalues and exact localization, with eigenfunctions compactly supported on only two spinorial sites. This extreme confinement surpasses the factorial decay characteristic of the classical scalar Stark model. For the general long-range Dirac model, we observe that the eigenvalues remain asymptotically close to the Stark ladder and prove that the corresponding eigenfunctions satisfy power-law localization estimates. Consequently, we establish power-law localization in the sense of finite moments of the position operator for the spinorial evolution. Our results demonstrate that deterministic Stark localization is robust and persists in genuinely matrix-valued lattice systems.

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math.SP 2026-06-18

Random potentials yield square-root cancellation in manifold eigenvalue bounds

by Jean-Claude Cuenin, Konstantin Merz +1 more

Random Schr\"odinger operators on manifolds and abstract bounds for multiplier-type operators

High-probability spectral inclusions for Anderson Schrödinger operators improve on deterministic estimates via a multiplier randomization pr

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We study random Schr\"odinger operators on closed Riemannian manifolds with Anderson-type potentials. We prove high-probability spectral inclusion bounds showing that eigenvalues remain close to those of the Laplacian, with deviations controlled by a norm of the potential coefficients. Compared with deterministic bounds, this yields a square-root cancellation gain. The proof is based on a general principle showing that randomisation improves operator norm bounds for multiplier-type operators, which we formulate in both discrete and continuous settings.

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math.SP 2026-06-18

Two spectra fix third-order operator coefficients

by Natalia P. Bondarenko

Third-order inverse spectral problem with the three-point boundary conditions

Uniqueness and existence follow from Weyl-Yurko matrix reconstruction for the given three-point boundaries.

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In this paper, we study a new inverse spectral problem that consists in the recovery of the third-order differential equation from two spectra corresponding to the boundary conditions $y(0) = y(1) = y(2) = 0$ and $y(0) = y'(0) = y(1) = 0$. The uniqueness and existence theorems for the solution are obtained. To prove the results, we treat the inverse problem using a general approach that reconstructs higher-order differential operators from the Weyl-Yurko matrix.

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math.DS 2026-06-18

Uniform spectral gap holds for cusped thin subgroups of SO(n,1)

by Pratyush Sarkar

Generalization of Selberg's 3/16 theorem for geometrically finite thin subgroups of operatorname{SO}(n, 1)

Proven when critical exponent exceeds 1/2 for n at least 3, covering cases with cusps that prior results left open.

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Let $\Gamma$ be a geometrically finite thin subgroup of an arithmetic lattice $\Gamma_0 < G := \operatorname{SO}(n, 1)$ and consider the congruence covers of $\Gamma \backslash G$. In the breakthrough work of Bourgain-Gamburd-Sarnak, the expansion machinery was used to establish a uniform spectral gap in the setting $(G, \Gamma_0) = (\operatorname{SL}_2(\mathbb{R}), \operatorname{SL}_2(\mathbb{Z}))$ when the critical exponent satisfies $\delta_\Gamma > \frac{1}{2}$. The main applications are affine sieve for $\Gamma$-orbits and uniform resonance-free half-planes for the resolvent of the Laplacian. These results were generalized in subsequent works by Mohammadi-Oh, Oh-Winter, the author, and Edwards-Oh. Yet, the region $\delta_\Gamma \in \bigl(\frac{1}{2}, n - 2\bigr]$ for $n \geq 3$ remains to be treated when there are cusps. The purpose of this paper is to fill in this gap in the literature. The difficulty lies in working with a countably infinite coding due to the presence of cusps. In particular, we incorporate new tools to prove the Zariski density and full trace field properties of the return trajectory subgroups.

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math.SP 2026-06-16

Point mass implies localized eigenfunctions in random networks

by Georgii Veprev

Localization of eigenfunctions in amenable unimodular random networks

Amenable unimodular networks have finite-support eigenfunctions with positive probability when their expected spectrum has atoms.

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For an amenable unimodular random rooted network, we show that the presence of a positive point mass in the expected spectral measure implies that, with positive probability, there exists an eigenfunction with finite support.

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math.LO 2026-06-12

Modal preorders keep two exact sources in CH23 SCI block

by Christopher Sorg

Finite-Query Collapse and Modal Exact Bases in the SCI Hierarchy

Raw finite-query reductions collapse spectral and pseudospectral sources; the CH23 geometric modality restores exactly two minimal bases.

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We study the exact-basis problem for Solvability Complexity Index (SCI) computational problem families through finite-query transports. A raw finite-query reduction permits arbitrary encodings and finite transcript reconstructions, with only a continuous output decoder. For the Colbrook-Hansen (CH23) singleton-window spectral/pseudospectral block, this raw preorder collapses the expected two-source structure: the diagonal exact spectral and fixed-$\varepsilon$ pseudospectral sources are raw- and continuous-finite-query equivalent, and, for computable $\varepsilon$ under the evaluation-name representations, TTE-finite-query equivalent, so the six-problem ambient is raw-principal. We then introduce modal finite-query preorders, whose admissibility conditions may restrict encodings, decoders, reconstructions, uniformity, and geometric naturality. We also characterize TTE finite-query transport as computable point transport with a uniform finite interface trace; after forgetting the trace this gives strong Weihrauch reducibility, and the implication is strict. Under a CH23 geometric modality generated by representation inclusions, unitary and graph relabelings, and neutral stabilizations, the same ambient has exactly two minimal exact sources. This gives a calibrated reformulation of the exact-basis problem: natural SCI families should be classified by modality-indexed exact bases and refinement maps, not by one raw preorder alone.

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math.AP 2026-06-12

Orthonormal spectral clusters gain log-improved bounds on curved manifolds

by Jean-Claude Cuenin, Ngoc Nhi Nguyen +1 more

Orthonormal Spectral Cluster Bounds on Manifolds with Nonpositive Curvature

Sharp estimates hold for windows of size 1 over log lambda via curvature-enabled kernel estimates.

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Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold with nonpositive sectional curvature. We prove sharp, logarithmically improved spectral cluster bounds for orthonormal systems in the supercritical range. More precisely, for spectral windows of size $(\log \lambda)^{-1}$, we obtain the orthonormal analogue of the logarithmically improved $L^q$ estimates of Hassell-Tacy. Our argument combines the universal orthonormal spectral cluster bounds of Frank-Sabin with B\'erard-type kernel estimates and a generalization of the Bourgain-Shao-Sogge-Yao multiplier estimate to the orthonormal setting.

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math.DG 2026-06-11

Fundamental gap lower bound scales as D^{-3} for large horoconvex domains

by Xianzhe Dai, John Ennis +2 more

A Large-Diameter Fundamental-Gap Lower Bound for Horoconvex Domains

The bound matches the known upper bound and holds in every dimension after reducing to a fixed-width radial model.

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We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature $-1$. The geometric part reduces large horoconvex domains to a fixed-width radial-height problem in all dimensions. The analytic part proves the needed radial-height theorem by comparing the low-energy Dirichlet form with a limiting angular operator on the sphere, while the radial complement is separated by a one-dimensional branch gap and endpoint Green estimates. The result gives the polynomial $D^{-3}$ scale matching the Nguyen--Stancu--Wei large-diameter upper bound.

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math.SP 2026-06-11

Pure point spectrum is generic for bounded potentials

by Artur Avila (Universität Zürich, IMPA) +1 more

Pure Point Spectrum is Generic

For comeager many sequences in the sup norm, the Schrödinger operator spectrum is a zero-weight Cantor set.

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We consider Schr\"odinger operators in $\ell^2(\mathbb{Z})$ with real-valued potentials in $\ell^\infty(\mathbb{Z})$ and show that the generic spectral type is pure point. More specifically, we show that for a generic bounded potential, the essential spectrum of the associated Schr\"odinger operator is a Cantor set and has zero weight with respect to all spectral measures.

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math.SP 2026-06-11

Perron-Frobenius theorem extended to complex-weighted networks

by Yu Tian, Mason A. Porter +1 more

Generalizing Perron--Frobenius theory and eigenvector-based centralities to networks with complex edge weights

Generalized versions define eigenvector centrality for quantum, circuit, and chemistry networks.

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A fundamental concept in linear algebra and its applications to network analysis is the Perron--Frobenius (PF) theorem, which underpins eigenvector-based centrality measures such as eigenvector centrality, PageRank, and hubs and authorities. By invoking the PF theorem, we know for strongly connected networks with positive edge weights that the eigenvector corresponding to the largest eigenvalue of the weight matrix yields a well-defined centrality measure (namely, eigenvector centrality). Traditional formulations of the PF theorem and associated centrality measures assume that networks have real-valued weights. However, many networks in areas such as quantum information, quantum chemistry, electrodynamics, and machine learning have complex-valued edge weights. In this paper, we study generalizations of the PF theorem to complex-valued matrices, establish connections between these generalizations, and propose generalized eigenvector-based centrality measures to analyzing node importances in networks with complex edge weights. We also prove results about the existence of complex-weighted networks that satisfy generalized PF properties and calculate associated centrality measures for several examples, which we draw from application areas such as electron transport, circuit analysis, mathematical chemistry, and communication networks.

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math.DG 2026-06-11

Supersymmetry duality pairs Dirichlet and Neumann spectra via S inversion

by Vicent Gimeno i Garcia, Paulo Henryque da Costa Silva

Dirichlet--Neumann duality for the Basic Spectrum of Riemannian Submersions: A Supersymmetric Perspective

Riemannian submersions with basic mean curvature fibers have their spectra related by inverting the fiber-volume function in supersymmetric

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This manuscript investigates the spectral geometry of Riemannian submersions whose fibers have a basic mean curvature. By restricting the Laplace--Beltrami operator to the space of basic functions, we reduce the spectral problem on $M$ to the spectral problem for a weighted Laplacian on the base manifold, where the weight is determined by the fiber-volume function $S$. We derive a summation formula for the reciprocal of the basic Dirichlet eigenvalues (Basel-type series). Furthermore, using the framework of Supersymmetric Quantum Mechanics (SUSYQM), we establish a supersym\-me\-tric duality relating the basic Dirichlet and Neumann spectra under the trans\-for\-ma\-tion $S \mapsto 1/S$.

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math.DG 2026-06-11

Sub-Laplacian determinant on SL(2,R) quotients factors via base surface

by Fabrice Baudoin

Sub-Riemannian Selberg Trace Formulae for Compact Quotients of SL(2,R) and Determinants of Sub-Laplacians

Trace formula reduces to Maass Laplacians, giving compact expression with relative Selberg product.

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We prove sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R). Using the Fourier decomposition along the SO(2)-fibers, we reduce the heat trace computation to the Selberg trace formula for Maass Laplacians on the hyperbolic plane. The resulting formula has an identity contribution and a hyperbolic contribution, the latter involving a character-dependent theta factor over closed geodesics. We then use this trace formula to compute the zeta-regularized determinant of the sub-Laplacian. The determinant formula is remarkably compact and is expressed in terms of a determinant depending only on the base hyperbolic surface and an explicit relative Selberg product.

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math.CO 2026-06-11

|ρ₂| ω ≤ m-2 for odd-order connected graphs

by Huiqiu Lin, Lianping Liu +2 more

Krahn--SzegH{o} type inequalities and nodal domain methods on graphs

Equality only for two cliques joined by edge or path; settles 2010 conjecture via nodal domains on graphs.

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We study discrete analogues of classical spectral geometric inequalities and extremal eigenvalue problems on graphs. The classical Krahn--Szeg\H{o} inequality states that, among bounded open subsets of $\mathbb{R}^n$ with fixed volume, the minimum of $\lambda_2(\Omega)$ is attained by the union of two congruent balls. Firstly, we establish a Krahn--Szeg\H{o} type inequality for trees. For trees with a fixed number of interior vertices and boundary leaves, we completely characterize the extremal structures that minimize the second Dirichlet eigenvalue. Secondly, we develop a nodal domain method for adjacency matrices. By proving an adjacency version of the nodal domain theorem for graphs, we obtain upper bounds for the second largest adjacency eigenvalue $\rho_2(G)$ of $G$ in given graph classes. These bounds imply some previous results. Finally, we settle the Aouchiche--Hansen conjecture (2010) on the second largest eigenvalue with given number of edges and clique number. We prove that for connected graphs $G$ of odd order $n \geq 5$, $|\rho_2| \cdot \omega \leq m-2$, with equality if and only if $G$ consists of two complete graphs of orders $\frac{n+1}{2}$ and $\frac{n-1}{2}$ joined by an edge or a path. For even $n \geq 2$, the quantity $|\rho_2| \cdot \omega - m$ is maximized exactly when $G$ is obtained by adding one edge between the two copies of $K_{n/2}$ by an edge. The core of the methods developed in this paper is to regard a connected graph as an internally disconnected graph with Dirichlet boundary condition. This perspective allows us to transfer nodal domain techniques from continuous spectral geometry to discrete settings and to obtain sharp extremal characterizations across diverse graph classes.

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math.PR 2026-06-10

CD manifold admits no Lipschitz map pushing Gaussian forward

by William Dudarov, Dan Mikulincer

Geometric obstructions to Lipschitz transport between weighted Hessian CD(kappa,infty) manifolds

The two-dimensional construction shows curvature bounds alone do not force existence of such transport maps.

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We construct a weighted Riemannian manifold $(\mathbb R^2,g,\mu)$ satisfying $\mathrm{CD}(1/2,\infty)$, the curvature-dimension condition, with the following property: if $\gamma$ denotes a centered Gaussian measure on $\mathbb R^2$, then there is no Lipschitz map $T:(\mathbb R^2,\|\cdot\|) \to (\mathbb R^2,g)$ satisfying $T_\#\gamma=\mu$. Building on this, we prove a Weyl-type asymptotic law for the eigenvalues of the weighted Laplacian $-\Delta_{g,\mu}$ and show that they are asymptotically negligible when compared to the eigenvalues of $-\Delta_{\gamma}$. These results give strong counterexamples to two questions of E. Milman and complement the recent counterexample of Aryan.

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math.SP 2026-06-10

Delta potential on equator yields non-invariant semiclassical measures

by Santiago Verdasco

Quantum Limits of the Laplacian perturbed along a geodesic on mathbb{S}²

Eigenfunction sequences can concentrate on one hemisphere, breaking the invariance that holds for bounded perturbations.

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This article studies the high-frequency behavior of eigenstates of perturbations of the Laplace-Beltrami operator on the two-sphere $\mathbb{S}^{2}$ by a measure supported on an equator. We are interested in understanding to what extent this behavior can be described in terms of the geodesic flow of the sphere. This is done by analyzing quantum limits and semiclassical measures of sequences of high-frequency eigenfunctions, which describe how their $L^2$-masses concentrate in phase space. When the Laplacian on $\mathbb{S}^{2}$ is perturbed by a bounded potential, it is known that the family of all possible semiclassical measures is contained in the set of positive measures on the unit cosphere bundle $S^*\mathbb{S}^{2}$ that are invariant under geodesic flow (with equality in the unperturbed case). In this article, we show that the presence of a singular delta potential on a closed geodesic results in the existence of sequences of eigenfunctions whose semiclassical measure is not invariant under geodesic flow. In particular, one can find a sequence of eigenfunctions whose energy asymptotically concentrates on the hemisphere bounded by the equator on which the potential is concentrated.

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math.SP 2026-06-10

Horoconvex gaps scale sharply as D to the minus three

by Sanghyun Park

Effective Angular Asymptotics and the Sharp D⁻³ Horoconvex Gap Scale

The Dirichlet fundamental gap on large horoconvex domains in hyperbolic space has leading constant set by a variational problem over sphere

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We prove first-band large-diameter asymptotics for the Dirichlet spectrum on horoconvex domains in real hyperbolic space. After Chebyshev centering, a divergent sequence compactifies to a horospherical support-envelope deficit \[V\] on \[\mathbb S^{n-1}\]. For graph domains \[r<R-V(\theta)\], the first band satisfies \[ \lambda_{j+1}=\alpha^2+\frac{\pi^2}{R^2} +\frac{2\pi^2}{R^3}\bigl(\eta_j(T_n+V)-b_0\bigr)+o(R^{-3}), \qquad j=0,1, \] where \[T_n\] is the nonlocal spherical operator with multiplier \[\psi(\ell+\alpha)-\psi(\alpha)\]. Consequently the horoconvex fundamental gap has the sharp \[D^{-3}\] polynomial scale, and the leading large-diameter constant is characterized by the compact variational formula \[ 16\pi^2\inf_{V\in\mathcal A_n} \bigl(\eta_1(T_n+V)-\eta_0(T_n+V)\bigr). \] Geodesic balls realize the polynomial scale, but an explicit admissible axial perturbation lowers the reduced leading-constant value at first order.

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math.RA 2026-06-09

Eigenvalue region of n-cycle stochastic matrices fully mapped

by Brecht Verbeken, Vincent Ginis

On the Spectral Region of n-Cycle Stochastic Matrices

For n≥3 the upper half is the image of K vertical angular sectors under the sine-ratio map, bounded by Jensen chords and algebraic arcs.

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For every $n$, we determine the complete eigenvalue region of the $n$-cycle stochastic family. For $n\ge 2$, write $A_n(\alpha)$ for the matrix indexed by $\mathbb Z/n\mathbb Z$ with $$ (A_n(\alpha))_{j,j}=\alpha_j,\qquad (A_n(\alpha))_{j,j+1}=1-\alpha_j,\qquad 0\le \alpha_j<1, $$ and all other entries zero, and set $C_n=\{A_n(\alpha):\alpha\in[0,1)^n\}$. Writing $\Sigma_n$ for the corresponding spectral union, the trivial cases are $\Sigma_1=\{1\}$ and $\Sigma_2=[-1,1]$. For $n\ge 3$, we give an explicit description of $\Sigma_n$ in angular coordinates $m=\mathrm{Arg}(\lambda)$ and $M=\mathrm{Arg}(\lambda-1)$. Under the map $$ \Lambda(m,M)=\frac{\sin M}{\sin(M-m)}e^{im}, $$ the upper half of $\Sigma_n$ is the image of a finite union of $K=\lfloor(n-1)/2\rfloor$ vertical angular sectors. Its exposed boundary is an alternating chain of Jensen chords, arising from the Jensen-equality lines $M=\phi_k$, and algebraic one-loop arcs joining the relevant roots of unity to $0$; the lower boundary is obtained by complex conjugation. The real spectral part is $[-1,1]$ for even $n$ and $(0,1]$ for odd $n$. The proof is independent of Karpelevich's theorem and reduces the two-monomial characteristic equation to sharp argument bounds on a simplex, obtained by Jensen, majorization, and finite visibility arguments.

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math.SP 2026-06-09

Reflectionlessness produces clock spacing in two-sided Jacobi matrices

by Benjamin Eichinger, Milivoje Lukić +1 more

Clock spacing for two-sided Jacobi matrices

A movable-point scaling limit shows the spacing follows from a local condition on the matrix coefficients rather than from absolute continui

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We study local eigenvalue spacing for finite truncations of a two-sided Jacobi matrix with two movable endpoints. In particular, we show that a suitable analog of clock spacing follows from a pointwise reflectionlessness condition. We obtain this as a consequence of a new scaling limit for Christoffel--Darboux kernels with a movable starting point. Without reflectionlessness, we obtain a new class of limit kernels, which combine distinct contributions from $\pm\infty$. We also show that clock spacing in the two-sided setting is a fragile phenomenon, which can be destroyed by the change of a single Jacobi coefficient; in particular, it is not merely a consequence of absolutely continuous spectrum.

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math.AP 2026-06-09

Two mechanisms control linear growth near Lamb-Chaplygin dipole

by Francesco Pio Numero, Paolo Ventura

Linear Stability of the Lamb-Chaplygin Dipole

Spectrum classification shows growth only from core circulation or zero-eigenvalue chains, implying dynamics may drift within the dipole fam

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We describe the linearized dynamics near the Lamb-Chaplygin dipole, a classical traveling solution of the two-dimensional Euler equations. Exploiting the Hamiltonian structure of the system together with its symmetries, we identify all possible sources of linear instability. For general perturbations in $L^1\cap L^p$, $p>2$, growth can occur only through two explicit mechanisms triggered by: {\rm (i)} a nonzero circulation on the core of the dipole, and {\rm (ii)} a nontrivial component along the generalized eigenvectors associated with the eigenvalue $0$. In particular, we completely classify the spectrum and the Jordan chains of the operator associated with the linear dynamics. Both mechanisms hint for a nonlinear dynamics that may drift along the symmetry-generated family of traveling dipoles without moving away from it.

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math.SP 2026-06-09

Star-graph Sturm-Liouville recovery is Lipschitz stable from Weyl vector

by Maria Kuznetsova

Uniform stability of recovering the Sturm-Liouville operator on a star-graph

Estimates depend only on potential norm bound, justifying numerical methods for the inverse problem.

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In the paper, we study the problem of recovering the Sturm-Liouville operator on a star-graph from the Weyl vector. It generalizes the problem of recovering the classical Sturm-Liouville operator on an interval from the Weyl function, and the problems of recovering from other spectral data can be reduced to this problem. The uniqueness and the constructive method for solving the problem under study were previously obtained by V.A. Yurko in the case of a tree (Inverse Problems, 2005). Here, we prove its uniform stability, which includes Lipschitz estimates with a constant depending only on the number bounding the norms of the potentials. Stability results are necessary for justifying the well-posedness of the problem statement, and they are important for developing numerical methods. As auxiliary results, we obtain the uniform stability of the direct problem, as well as the uniform stability of the partial derivatives of the transmutation operator kernel related to the classical Sturm-Liouville operator.

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math.SP 2026-06-09

Generic potentials give Schrödinger operators maximal resonance density

by Travis Cunningham

Schroedinger operators with generic potentials achieve maximal resonance density

The integrated counting function hits the optimal asymptotic upper bound for almost every compactly supported potential.

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We show that for a generic real or complex-valued compactly supported potential, the corresponding Schroedinger operator achieves maximal resonance density, in the sense that its integrated resonance counting function achieves the optimal asymptotic upper bound. For odd dimensions this follows from results of Dinh-Vu once we adapt an argument of Christiansen Hislop. The proof for even dimensions constitutes the bulk of the paper, and we prove several new results on resonances which have analogues in the odd dimensional case. This includes a sharp upper bound on the integrated resonance counting function for any compactly support potential, a proof that the characteristic function of a ball has resonance counting function which achieves the optimal upper bound, and an even-dimensional analogue of the result of Dinh-Vu on asymptotics of the resonance counting functions for complements of pluripolar subsets of analytic families of potentials. We use the characterization of resonances as zeros of certain Fredholm determinant functions related to the scattering matrix, allowing us to apply techniques and results from the theories of one and several complex variables. Our proof that the characteristic function of a ball has counting function achieving the optimal upper bound uses the uniform asymptotics of Bessel functions and follows ideas of Zworski, Christiansen-Hislop, and Dinh-Vu.

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math.SP 2026-06-09

Near isospectrality implies full isospectrality for symmetric quotients

by Sudhir Pujahari, Punya Plaban Satpathy

Near Isospectrality and Spectral Rigidity for Compact Locally Symmetric Manifolds

Compact quotients of fixed symmetric spaces of nonpositive curvature cannot differ in only finitely many eigenvalues.

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The inverse spectral problem asks to what extent the Laplace--Beltrami spectrum determines the geometry of a Riemannian manifold. We study a natural weakening, called \emph{near isospectrality}, in which the spectra of two compact manifolds agree outside a finite set, counted with multiplicity. We prove that for compact quotients of a fixed simply connected symmetric space of nonpositive sectional curvature, near isospectrality already forces full isospectrality. We then extend this rigidity to a broad collection of compact quotients of irreducible symmetric spaces of noncompact type. In this larger setting, near isospectrality determines enough heat invariants to identify the universal cover within the class under consideration, and the fixed-cover rigidity result then implies full isospectrality. Thus, within the class studied here, eventual agreement of the Laplace spectrum already forces complete spectral agreement.

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math.AP 2026-06-08

Toral eigenfunctions meet sharp L2 restriction bounds at high codimension

by Cheng Zhang, Zhifei Zhu

Restriction estimates for toral eigenfunctions and lattice points in spherical regions

This settles the conjecture on submanifolds of large codimension and gives new lattice point estimates.

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We establish new $L^2$ restriction estimates for toral eigenfunctions. These estimates are sharp in certain cases, and thus prove a conjecture of Huang-Zhang for smooth submanifolds of large codimension. In particular, they provide new progress toward a conjecture of Bourgain-Rudnick. The proof combines a slicing and packing method with the approximation of the discrete spherical multiplier by Magyar-Stein-Wainger and Magyar.

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math.AP 2026-06-08

Equality holds only for homothetic domains in Monge-Ampère Brunn-Minkowski

by Nam Q. Le

On the equality case in the Brunn-Minkowski inequality for the Monge-Amp\`ere eigenvalue

Characterization for general bounded convex domains completes resolution of Salani's 2005 question.

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We characterize the equality case in the Brunn-Minkowski inequality for the Monge-Amp\`ere eigenvalue of general bounded convex domains. This together with the author's previous work completely resolves a question raised by Salani (A Brunn-Minkowski inequality for the Monge-Amp\`ere eigenvalue. Adv. Math. 194 (2005)).

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math.SP 2026-06-08

Ball minimizes sum of reciprocal Neumann eigenvalues

by Yixin He, Yanyang Li +1 more

A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues

For smooth domains of fixed volume the ball is the unique minimizer of the sum of the first m reciprocal eigenvalues.

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We prove the Ashbaugh--Benguria conjecture for bounded domains with smooth boundary in $\mathbb R^m$. More precisely, among all smooth bounded domains of fixed volume, the ball minimizes the sum of the reciprocals of the first $m$ nonzero Neumann eigenvalues. Equality is attained precisely by balls.

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math.SP 2026-06-08

Hankel operator eigenfunctions mirror Schrödinger solutions

by Alexander Pushnitski

Eigenfunctions of positive integral Hankel operators

The integral equation Hf=Ef on the positive semi-axis displays the same qualitative properties as the one-dimensional Schrödinger equation f

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We consider bounded positive semi-definite Hankel operators $H$, realised as integral operators on the positive semi-axis. For each value of $E$, not necessarily in the spectrum of $H$, we analyse solutions $f$ of the eigenvalue equation $Hf=Ef$, understood as an integral equation on the semi-axis. Our analysis reveals strong analogies with properties of solutions of the one-dimensional Schr\"odinger equation.

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math.CO 2026-06-08

New constant gives sharp bounds on second Laplacian gap

by Lies Beers, Raffaella Mulas +1 more

Cheeger-type inequalities for the second largest spectral gap from 1 of the normalized Laplacian

Two-step random-walk escape probability controls the gap from 1 in the normalized Laplacian spectrum with classical Cheeger sharpness.

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We study the second largest spectral gap from $1$ of the normalized Laplacian of a graph, a quantity that appears in the literature in connection with random walks, expander graphs, and Ramanujan graphs. We relate it to the classical Cheeger and dual Cheeger constants, and we introduce a new Cheeger-type constant admitting a probabilistic interpretation in terms of two-step random walks. For this constant, we establish sharp inequalities analogous to the classical Cheeger inequalities.

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math.DG 2026-06-08

One eigenvalue condition forces a graph to be a hypercube

by Yanlong Ding, Shiping Liu +1 more

Optimal spectral rigidity of the hypercube via Bakry--\'Emery curvature

Under a positive Bakry-Émery curvature bound the (Δ-1)th Laplacian eigenvalue equal to K identifies the hypercube among unweighted graphs.

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Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. We establish the following spectral rigidity theorem. Let $G$ be a finite, connected, simple, unweighted graph with Bakry--\'Emery curvature bounded below by $K>0$. Denote by $\Delta$ the maximum degree of $G$, and let $0=\lambda_0<\lambda_1\leq\cdots$ be the eigenvalues of the non-normalized Laplacian. Then $$ \lambda_{\Delta-1}=K \quad\Longrightarrow\quad G\cong H_\Delta, $$ where $H_\Delta$ is the $\Delta$-dimensional hypercube graph. Thus, in the unweighted setting, the multiplicity condition $\lambda_{\Delta}=K$ appearing in the hypercube rigidity theorem of Liu, M\"unch, and Peyerimhoff can be weakened to $\lambda_{\Delta-1}=K$. This improvement is optimal. The restriction to unweighted graphs is essential: the strengthened rigidity statement fails in the weighted setting. Our argument is built upon an interplay between the global spectral embedding induced by the first eigenspace and a local analysis of curvature matrices.

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math-ph 2026-06-08

King functions from shifted Gaussians span radial velocity space densely

by Yanpeng Wang, Zhe Gao

King Function for Shifted Gaussian: Laguerre Structure, Spectral Theory and Density

Unitary map to the radial Schrödinger operator places real-parameter King functions in the resolvent set and yields an approximation basis.

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We study King function arising as radial kernels in the laboratory-frame spherical harmonic expansion of shifted Gaussian distributions. We first clarify their relation with the co-moving Laguerre hierarchy by means of a King--Laguerre expansion. We then derive the King differential equation and show that the associated self-adjoint operator in a Gaussian-weighted Hilbert space is unitarily equivalent to the free radial Schr\"odinger operator on the half-line. This yields the spectral representation and generalized eigenfunction. Finally, we prove that real-parameter King function, lies in the resolvent set, form a dense non-orthogonal system in a natural radial velocity space, providing an approximation-theoretic basis for King mixture representations. Weighted $L^1$-integrability criteria and closed-form moment formulas are also derived, justifying the normalization of King function.

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math-ph 2026-06-08

Magnetic Cheeger constant equals cycle flux distance to lattice

by Björn Dahlke

Flux-explicit Cheeger bounds for magnetic Laplacians on compact metric graphs

Frustration index simplifies to ℓ¹ distance, giving explicit bounds on spectrum bottom and heat decay without subgraph search.

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Let (\Gamma) be a finite compact connected metric graph and let (A\in L^\infty(\Gamma)) be a real magnetic potential. The magnetic Laplacian (H_A) with standard vertex conditions is defined by the closed quadratic form [ q_A[u]=\sum_e\int_e |(-i\partial_x-A_e)u_e|^2,dx. ] A magnetic Cheeger constant is introduced by adding to the usual boundary term the frustration index of the potential on subgraphs. The first point of the paper is that, on a metric graph, this frustration index is exactly a finite dimensional (\ell^1) flux distance determined by the periods of (A) on cycles. Consequently the Cheeger constant can be written directly in terms of Aharonov Bohm fluxes. We prove a Cheeger type lower bound for the bottom of the spectrum and derive the corresponding explicit lower estimate in terms of the distance of the global cycle flux vector from the integral flux lattice. The estimate also gives (L^2) decay bounds for the magnetic heat semigroup and for the magnetic energy. The constants are not asserted to be sharp; the emphasis is on the flux dependence and on the self contained metric graph formulation.

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math.CA 2026-06-05

Sparse square-summable data converges under SU(1,1) and SU(2) nonlinear Fourier transform

by Sergey A. Denisov

Convergence of sparse square-summable NLFT

The same conditions produce asymptotics for the associated orthogonal polynomials on the unit circle.

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We study the convergence of SU(1,1) and SU(2) nonlinear Fourier transform with sparse square-summable data. The asymptotics of the associated polynomials orthogonal on the unit circle is obtained as a corollary.

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math.ST 2026-06-05

Inference impossible above estimation limit for subspaces

by Joshua Agterberg

Statistically and Computationally Optimal Estimation and Inference of Common Subspaces

Shared subspace estimation succeeds at lower SNR than adaptive confidence intervals, which remain impossible until a higher threshold.

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Given multiple data matrices, many problems in statistics and data science rely on estimating a common subspace that captures certain structure shared by all the data matrices. In this paper we investigate the statistical and computational limits for the common subspace model in which one observes a collection of symmetric low-rank matrices perturbed by noise, where each low-rank matrix shares the same common subspace. Our main results identify several regimes of the signal-to-noise ratio (SNR) such that estimation and inference are statistically or computationally optimal, and we refer to these regimes as weak SNR, moderate SNR, strong estimation SNR, and strong inference SNR. First, we propose an estimator based on projected gradient descent initialized via spectral sum of squares and show that it achieves the optimal $\sin\Theta$ error rate under strong estimation SNR. These results are complemented by both statistical and computational lower bounds identifying the weak and moderate estimation SNR regimes. Next, we turn to statistical inference for the $\sin\Theta$ distance itself, and we show that our estimator has an asymptotically Gaussian distribution in the strong inference SNR regime. Based on this limiting result we propose confidence intervals and show that they are adaptively minimax optimal in the strong inference SNR regime, where adaptivity is measured in terms of the SNR. Finally, we show that adaptive confidence intervals are information-theoretically impossible below the strong inference SNR regime. Consequently, our results unveil a novel phenomenon: despite the SNR being ``above'' the computational limit for estimation, adaptive statistical inference may still be information-theoretically impossible.

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math.DG 2026-06-05

Second derivative of η germ separates lens spaces

by Sanchita Sharma

Second-Jet Equivariant η Separations on Lens Spaces

For L(25,4) versus L(25,9) the normalized value is -6080 while ordinary η and first derivative agree.

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Lens spaces are useful test examples in spectral geometry because their spin Dirac eigenspaces admit explicit congruence descriptions. We use these descriptions to study equivariant $\eta$ invariants for three-dimensional lens spaces with the round metric and the standard coordinate-torus action, retaining the spin-Fourier character of each eigenspace rather than only the ordinary scalar $\eta$ value. For the square family $L(\ell^2,\ell-1)$ and $L(\ell^2,2\ell-1)$, with $\ell\geq 5$ odd, we obtain a residual-circle equivariant $\eta$ separation: the ordinary $\eta$ values agree, and the first derivative of the residual $\eta$ germ vanishes by symmetry, but the second derivative is nonzero. For $L(25,4)$ versus $L(25,9)$, the normalized second derivative is $-6080$. Thus, the residual-circle equivariant $\eta$ germ detects a distinction invisible to the ordinary $\eta$ invariant. The calculation uses spin-Fourier residues directly; perturbative Hessian signs serve only as motivation and are not part of the invariant.

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math.SP 2026-06-05

Brush graphs set exact positions for n Laplacian spectral gaps

by Andrii Khrabustovskyi, Anna Muranova

Periodic discrete graphs with prescribed spectrum

An infinite chain with n pendants per vertex has gap endpoints and upper spectrum edge controlled by solving for edge weights.

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We construct a periodic weighted graph whose discrete Laplacian has a spectrum with precisely $n$ gaps. Moreover, we show that by an appropriate choice of the weights, the endpoints of these gaps, as well as the upper edge of the spectrum, attain the prescribed values. The underlying graph has a brush-like geometry: it consists of an infinite chain of vertices, each of which is connected to $n$ additional pendant vertices by extra edges. Semi-explicit formulae for the weight coefficients are provided: some of the coefficients are determined explicitly, while others are given as roots of an explicitly determined polynomial.

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math.AP 2026-06-05

Superellipse damping yields explicit wave decay lower bounds

by B. Achammer, Perry Kleinhenz

Optimal decay for waves damped by superellipses

The rates depend on the superellipse exponent and the polynomial power of the damping, and are optimal for some parameter choices.

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Energy decay rates for solutions of the damped wave equation on the torus are known to be influenced by the geometry of the damped set and the growth properties of the damping. In this paper we produce lower bounds on energy decay rates for a class of damping which are positive on a superellipse and grow polynomially like the distance to the boundary of the superellipse. The energy decay rates we obtain depend explicitly on the exponent used to define the superellipse and the polynomial power. We show these rates are sometimes optimal. The proof adapts quasimodes from $y$-invariant damping using a simplification of the usual normal form argument.

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math.SP 2026-06-05

Trivially perfect graphs forbid eigenvalues in [√8-4,0] except -1,0

by Cristian M. Conde, Ezequiel Dratman +1 more

A Sharp Forbidden Interval for the Nontrivial Adjacency Eigenvalues of Trivially Perfect Graphs

Spectrum intersection with the interval contains only -1 and 0, and the interval is sharp at both ends.

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We prove a sharp forbidden interval for the nontrivial adjacency eigenvalues of trivially perfect graphs. More precisely, we show that if $G$ is a trivially perfect graph, then $\operatorname{Spec}(G)\cap [\sqrt{8}-4,0]\subseteq \{-1,0\}$. Moreover, we prove that the interval is best possible at both endpoints: there are connected trivially perfect graphs with eigenvalues arbitrarily close to $\sqrt{8}-4$ from below, and connected trivially perfect graphs with positive eigenvalues converging to $0$.

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cs.LG 2026-06-04

DeepMDMD enforces exact Koopman product rule on learned latent partitions

by Kelan Gray, Finlay Brown +2 more

Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning

Alternating multiplicative updates with clustering produces compact dictionaries that stay stable in noisy high-dimensional flows.

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Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy $Re=20,000$ lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

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math.AP 2026-06-04

Pleijel nodal bound extends to degenerate elliptic operators

by Rupert L. Frank, Bernard Helffer

Pleijel's theorem for a class of degenerate elliptic operators

Eigenfunctions of Baouendi-Grushin and boundary-degenerate cases obey the same asymptotic limit on sign changes as the Laplacian.

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We prove an asymptotic upper bound on the number of nodal domains of eigenfunctions of a class of degenerate elliptic operators. Our proof yields the same constant as in Pleijel's bound for the Dirichlet Laplacian. The operators considered include the Baouendi-Grushin operator and operators with ellipticity degenerating on the boundary.

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math.SP 2026-06-04

Explicit bound on exponential remainder in polygon heat trace

by Gustav M{aa}rdby

Beyond Three Terms: Exponential Bounds in the Neumann Heat Trace of Polygons

Locality principles for the heat kernel give a concrete estimate for how small the error is after three terms.

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We study the short-time asymptotic behavior of the heat trace associated with the Neumann Laplacian on polygonal domains in the plane. By establishing locality principles for the heat kernel near corners, edges, and the interior, we approximate the heat kernel on the polygon by model heat kernels defined on infinite sectors, half-planes, and the full plane, respectively. Although it is known that the Neumann heat trace of polygons admits a three-term asymptotic expansion followed by an exponentially small remainder, an explicit estimate for the exponent in this remainder term is not known. In this article, we provide such an estimate. We also discuss whether the exponent is sharp, and how it relates to known results. Finally, we discuss issues that arise when trying to extend the results to Robin boundary conditions.

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math.SP 2026-06-04

Eigenvalues of extensions read from Q-function and parameter

by Annemarie Luger, Jakob Reiffenstein

On eigenvalues of self-adjoint extensions for defect larger than one

The characterisation covers arbitrary defect and works in both Hilbert and Pontryagin spaces.

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Self-adjoint extensions of a symmetric operator are parametrised by Krein's formula, in which the $Q$-function interacts with another analytic function (the parameter). We obtain a characterisation of the eigenvalues, isolated or not, of a given self-adjoint extension in terms of these two functions. The setting is highly general, covering symmetric operators with arbitrary defect in a Hilbert or Pontryagin space. Of independent interest is our newly developed tool, the \emph{generalised value} of a generalised Nevanlinna function, for which we give both a function-theoretic and an operator-theoretic description.

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