pith. sign in

arxiv: 2606.09358 · v2 · pith:R4CACUK5new · submitted 2026-06-08 · 🧮 math.SP · math.AP

Schroedinger operators with generic potentials achieve maximal resonance density

Travis Cunningham This is my paper

Pith reviewed 2026-06-27 14:10 UTC · model grok-4.3

classification 🧮 math.SP math.AP
keywords Schrödinger operatorsresonancesresonance counting functionFredholm determinantsscattering matrixgeneric potentialscompact supportcomplex variables
0
0 comments X

The pith

For generic compactly supported potentials, Schrödinger operators achieve maximal resonance density.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that generic real or complex-valued compactly supported potentials make the associated Schrödinger operator attain the highest possible resonance density. This means the integrated resonance counting function meets the best known asymptotic upper bound. In odd dimensions the result adapts existing work, while even dimensions require new arguments based on complex analysis. A reader cares because this indicates that maximal density is the typical behavior for these operators rather than a special case.

Core claim

For a generic compactly supported potential the resonances of the Schrödinger operator, identified as zeros of Fredholm determinant functions from the scattering matrix, satisfy that their integrated counting function achieves the optimal asymptotic upper bound.

What carries the argument

Fredholm determinant functions associated to the scattering matrix, whose zeros correspond to resonances, to which results from one and several complex variables are applied.

If this is right

  • In odd dimensions the result follows from Dinh-Vu after adapting an argument of Christiansen and Hislop.
  • A sharp upper bound on the integrated resonance counting function holds for any compactly supported potential in even dimensions.
  • The characteristic function of a ball has a resonance counting function achieving the optimal upper bound.
  • An even-dimensional version of the Dinh-Vu result holds for complements of pluripolar subsets of analytic families of potentials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Maximal resonance density may hold for a dense set of potentials in appropriate topologies.
  • The techniques could apply to other operators where resonances are zeros of analytic functions.
  • Non-generic potentials might exhibit lower density but can be perturbed to achieve maximal density.
  • Similar generic results might exist for resonance problems in higher-order or systems of equations.

Load-bearing premise

Resonances correspond to the zeros of Fredholm determinant functions related to the scattering matrix.

What would settle it

A concrete compactly supported potential, real or complex, for which the integrated resonance counting function stays strictly below the optimal asymptotic upper bound in some dimension.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that for a generic real or complex-valued compactly supported potential, the associated Schrödinger operator achieves maximal resonance density: its integrated resonance counting function attains the optimal asymptotic upper bound. Odd-dimensional cases adapt Dinh-Vu via Christiansen-Hislop; even-dimensional cases form the bulk of the work and include a sharp upper bound for arbitrary compactly supported potentials, equality for the characteristic function of a ball (via uniform Bessel asymptotics), and an even-dimensional analogue of the Dinh-Vu pluripolar-set result. Resonances are characterized as zeros of Fredholm determinants tied to the scattering matrix, permitting application of one- and several-complex-variable techniques.

Significance. If the even-dimensional analytic results hold, the paper would establish a substantial extension of resonance-density results to generic potentials in all dimensions, together with new tools (sharp bounds, ball example, determinant properties) that may apply more broadly in scattering theory. The explicit use of complex-analysis methods on the scattering-matrix determinants is a methodological strength.

major comments (2)
  1. [even-dimensional section (bulk of paper)] Even-dimensional Dinh-Vu analogue (bulk of the paper, as described in the abstract): the new results on the Fredholm determinant must explicitly verify holomorphy of finite order in the relevant half-plane and confirm that zero-counting satisfies the precise growth hypotheses needed for the pluripolar-set conclusion; differences in phase factors and possible logarithmic terms relative to odd dimensions are not addressed in the provided description and are load-bearing for transferring the Dinh-Vu argument.
  2. [section on ball example and sharp upper bound] Sharp upper bound for arbitrary compactly supported potentials and equality for the ball (abstract and the section proving the ball example): the error estimates arising from uniform Bessel asymptotics must be shown to be strong enough to attain the exact optimal bound without residual logarithmic factors; this feeds directly into the generic statement.
minor comments (2)
  1. The introduction should state the precise form of the optimal upper bound and the definition of 'maximal resonance density' with a forward reference to the relevant theorem.
  2. Ensure that all citations to Dinh-Vu, Christiansen-Hislop, and Zworski are complete and that any new determinant results are clearly distinguished from prior work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help strengthen the exposition of our results on resonance density for Schrödinger operators. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [even-dimensional section (bulk of paper)] Even-dimensional Dinh-Vu analogue (bulk of the paper, as described in the abstract): the new results on the Fredholm determinant must explicitly verify holomorphy of finite order in the relevant half-plane and confirm that zero-counting satisfies the precise growth hypotheses needed for the pluripolar-set conclusion; differences in phase factors and possible logarithmic terms relative to odd dimensions are not addressed in the provided description and are load-bearing for transferring the Dinh-Vu argument.

    Authors: The even-dimensional analysis establishes holomorphy of finite order for the relevant Fredholm determinants in the half-plane via the analytic continuation properties of the scattering matrix and associated determinant estimates. The zero-counting function is shown to meet the precise growth hypotheses of the Dinh-Vu theorem (including the necessary order and type conditions), with explicit adjustments for the phase factors that differ from the odd-dimensional case and for any logarithmic contributions arising in the even-dimensional scattering theory. These verifications are load-bearing and appear in the proofs of the new results on the determinants and the pluripolar-set conclusion. To address the referee's concern about explicitness, we will add a clarifying paragraph or subsection summarizing these verifications and their relation to the odd-dimensional case. revision: yes

  2. Referee: [section on ball example and sharp upper bound] Sharp upper bound for arbitrary compactly supported potentials and equality for the ball (abstract and the section proving the ball example): the error estimates arising from uniform Bessel asymptotics must be shown to be strong enough to attain the exact optimal bound without residual logarithmic factors; this feeds directly into the generic statement.

    Authors: The sharp upper bound for arbitrary compactly supported potentials is obtained from the Fredholm determinant representation, and the ball example uses uniform Bessel asymptotics to achieve equality with the optimal bound. The error terms in these asymptotics are controlled at a rate sufficient to exclude residual logarithmic factors in the integrated counting function (yielding a clean leading-term asymptotic). This is essential for the subsequent generic statement and is verified directly in the ball computation. We will revise the relevant section to display the error estimates more prominently and confirm the absence of logarithmic remainders. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorems and original proofs

full rationale

The paper adapts Dinh-Vu and Christiansen-Hislop results for odd dimensions and supplies independent new proofs for even dimensions, including a sharp upper bound, the ball achieving equality via Bessel asymptotics, and an even-dimensional pluripolar-set analogue. Resonances are characterized as zeros of Fredholm determinants tied to the scattering matrix, permitting direct application of one- and several-complex-variables results. No quoted step reduces a claimed prediction or uniqueness statement to a fitted input, self-definition, or load-bearing self-citation chain; all central claims rest on externally verifiable analytic properties and original analysis rather than circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard theorems from complex analysis and Fredholm theory plus the cited external results of Dinh-Vu; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Resonances of the Schrödinger operator are the zeros of certain Fredholm determinant functions associated to the scattering matrix.
    Invoked to transfer the problem into the setting of several complex variables.
  • standard math Uniform asymptotics for Bessel functions hold in the required regime.
    Used to verify that the characteristic function of a ball achieves the optimal counting bound.

pith-pipeline@v0.9.1-grok · 5739 in / 1335 out tokens · 24683 ms · 2026-06-27T14:10:39.748574+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

31 extracted references

  1. [1]

    Chen,A sub-logarithmic lower bound for resonance counting function in two-dimensional potential scattering, Rep

    L.-H. Chen,A sub-logarithmic lower bound for resonance counting function in two-dimensional potential scattering, Rep. Math. Phys.65(2010), no. 2, 157–164

    2010

  2. [2]

    Christiansen,Several complex variables and the distribution of resonances in potential scat- tering, Comm

    T. Christiansen,Several complex variables and the distribution of resonances in potential scat- tering, Comm. Math. Phys.259(2005), no. 3, 711–728

    2005

  3. [3]

    2, 313–323

    ,Schrödinger operators with complex-valued potentials and no resonances, Duke Math J.133(2006), no. 2, 313–323

    2006

  4. [4]

    5, 961 – 982

    ,Schrödinger operators and the distribution of resonances in sectors, Analysis & PDE 5(2012), no. 5, 961 – 982

    2012

  5. [5]

    ,Resonant rigidity for Schrödinger operators in even dimensions, Annales Henri Poincaré20(2017), 1543–1582

    2017

  6. [6]

    Christiansen and T

    T. Christiansen and T. Cunningham,Singularities and asymptotic distribution of resonances for Schrödinger operators in one dimension, Asymptotic Analysis143(2025), no. 1, 619–640

    2025

  7. [7]

    Christiansen and P.D

    T. Christiansen and P.D. Hislop,The resonance counting function for Schrödinger operators with generic potentials, Math. Res. Lett.12(2005), no. 5-6, 821–826

    2005

  8. [8]

    ,Maximal order of growth for the resonance counting functions for generic potentials in even dimensions, Indiana Univ. Math. J.59(2010), no. 2, 621–660

    2010

  9. [9]

    Dinh and V.-A

    T.-C. Dinh and V.-A. Nguyên,Distribution of scattering resonances for generic Schrödinger operators, J. Funct. Anal.278(2020), no. 10, 1–26

    2020

  10. [10]

    Dinh and D.-V

    T.-C. Dinh and D.-V. Vu,Asymptotic number of scattering resonances for generic Schrodinger operators, Comm. Math. Phys.326(2014), no. 1, 185–208

    2014

  11. [11]

    Dyatlov and M

    S. Dyatlov and M. Zworski,Mathematical Theory of Scattering Resonances, Mathematical Surveys and Monographs, vol. 200, American Mathematical Society, 2019

    2019

  12. [12]

    Froese,Asymptotic distribution of resonances in one dimension, J

    R. Froese,Asymptotic distribution of resonances in one dimension, J. Differential Equations 137(1997), no. 2, 251–272

    1997

  13. [13]

    ,Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions, Canad. J. Math50(1998), no. 3, 538–546

    1998

  14. [14]

    Intissar,A polynomial bound on the number of scattering poles for a potential in even- dimensional spacesR n, Comm

    A. Intissar,A polynomial bound on the number of scattering poles for a potential in even- dimensional spacesR n, Comm. Partial Differential Equations11(1986), no. 3, 367–396

    1986

  15. [15]

    Lelong and L

    P. Lelong and L. Gruman,Entire Functions of Several Complex Variables, Grundlehren der Mathematischen Wissenschaften, vol. 282, Springer, 1986. 20

    1986

  16. [16]

    Olver,The asymptotic expansion for Bessel functions of large order, Phil

    F.W.J. Olver,The asymptotic expansion for Bessel functions of large order, Phil. Trans. Royal Soc.247(1954), 328–368

    1954

  17. [17]

    ,Asymptotics and Special Functions, Academic Press, 1974

    1974

  18. [18]

    Ransford,Potential Theory in the Complex Plane, Cambridge University Press, 1995

    T. Ransford,Potential Theory in the Complex Plane, Cambridge University Press, 1995

    1995

  19. [19]

    Regge,Analytic properties of the scattering matrix, Nuovo Cimento8(1958), no

    T. Regge,Analytic properties of the scattering matrix, Nuovo Cimento8(1958), no. 10, 671– 679

    1958

  20. [20]

    Sá Barreto,Lower Bounds for the number of resonances in even-dimensional potential scat- tering, J

    A. Sá Barreto,Lower Bounds for the number of resonances in even-dimensional potential scat- tering, J. Funct. Anal.169(1999), no. 3, 314–323

    1999

  21. [21]

    Simon,Operators with singular continuous spectrum I

    B. Simon,Operators with singular continuous spectrum I. general operators, Ann. Math141 (1995), 111–145

    1995

  22. [22]

    ,Resonances in one dimension and Fredholm determinants, J. Funct. Anal.178(2000), no. 2, 396–420

    2000

  23. [23]

    Smith and M

    H.F. Smith and M. Zworski,Heat traces and existence of scattering resonances for bounded potentials, Ann. Institute Fourier66(2016), no. 2, 455–475 (English, with English and French summaries)

    2016

  24. [24]

    Stefanov,Sharp upper bounds on the number of the scattering poles, J

    P. Stefanov,Sharp upper bounds on the number of the scattering poles, J. Funct. Anal.231 (2006), no. 1, 111–142

    2006

  25. [25]

    Tang and A

    S.H. Tang and A. Sá Barreto,Existence of resonances in even dimensional potential scattering, Comm. Partial Differential Equations25(2000), no. 5-6, 1143–1151

    2000

  26. [26]

    Vodev,Sharp bounds on the number of scattering poles in even-dimensional spaces, Duke Math

    G. Vodev,Sharp bounds on the number of scattering poles in even-dimensional spaces, Duke Math. J.74(1994), no. 1, 1–17

    1994

  27. [27]

    Nachr.170(1994), 287–297

    ,Sharp bounds on the number of scattering poles in the two-dimensional case, Math. Nachr.170(1994), 287–297

    1994

  28. [28]

    M Zworski,Distribution of poles for scattering on the real line, J. Funct. Anal.73(1987), no. 2, 277–296

    1987

  29. [29]

    J.59(1989), no

    ,Sharp polynonial bounds on the number of scattering poles, Duke Math. J.59(1989), no. 2, 311–323

    1989

  30. [30]

    ,Sharp polynonial bounds on the number of scattering poles for radial potentials, J. Funct. Anal.82(1989), no. 2, 370–403

    1989

  31. [31]

    ,Mathematical study of scattering resonances, Bull. Math. Sci.7(2017), no. 1, 1–85. Email address: travisdcunningham@gmail.com 21

    2017