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arxiv: 2606.00650 · v2 · pith:MQVHW3PXnew · submitted 2026-05-30 · 🧮 math.SP · math-ph· math.MP

Eigenfunction correlators under power-law SULE and localization for lattice operators

M. Aloisio , C. R. de Oliveira , R. Matos , D. Oliveira , M. Pigossi This is my paper

Pith reviewed 2026-06-30 11:31 UTC · model grok-4.3

classification 🧮 math.SP math-phmath.MP
keywords power-law SULEeigenfunction correlatorslocalization centerslattice operatorsdynamical localizationStark potentialsspectral gaps
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The pith

Power-law SULE forces equidistribution of localization centers and quantitative bounds on eigenfunction correlators for lattice operators.

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

The paper establishes a deterministic framework in which the assumption of power-law semi-uniform localization of eigenfunctions imposes geometric constraints, such as equidistribution of the centers, together with explicit upper bounds on eigenfunction correlators. The same correlator bounds can be reversed to recover the power-law SULE property itself. As a dynamical consequence the position operator has only finite q-moments of localization for q up to a definite threshold. The framework is applied directly to long-range operators with sublinear Stark potentials whose gaps collapse asymptotically, without any perturbative expansion.

Core claim

Power-law SULE yields geometric constraints on localization centers (such as their equidistribution) and quantitative bounds on eigenfunction correlators. As a dynamical consequence we obtain power-law localization in the sense of finite q-moments (up to a certain power q) of the position operator. Conversely, suitable bounds on eigenfunction correlators imply a corresponding form of power-law SULE, establishing a close connection between these notions.

What carries the argument

power-law semi-uniform localization of eigenfunctions (SULE), which supplies polynomial decay away from each localization center and thereby controls both geometry of the centers and size of correlators

If this is right

  • Localization centers must satisfy geometric constraints including equidistribution.
  • Eigenfunction correlators obey explicit quantitative upper bounds derived from the SULE rate.
  • The position operator exhibits power-law localization through finite q-moments up to a definite threshold.
  • Bounds on eigenfunction correlators are equivalent to the power-law SULE property.
  • The same conclusions apply to long-range operators with sublinear Stark potentials without perturbative assumptions.

Where Pith is reading between the lines

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

  • The geometric character of the argument suggests that spatial arrangement of centers, rather than randomness, can be the primary driver of localization features in deterministic models.
  • The equivalence between SULE and correlator bounds may provide a route to classify the transition between exponential and polynomial localization regimes on lattices.
  • The framework could be tested on other deterministic operators whose spectral gaps collapse at a controlled rate.

Load-bearing premise

The lattice operators under study are assumed to satisfy a power-law form of semi-uniform localization of eigenfunctions.

What would settle it

A concrete lattice operator obeying power-law SULE whose localization centers fail to equidistribute or whose eigenfunction correlators exceed the derived quantitative bounds.

read the original abstract

We develop a deterministic framework showing that a power-law form of semi-uniform localization of eigenfunctions (SULE) imposes strong structural constraints on lattice operators, with consequences of both spectral and dynamical nature. For instance, as spectral consequences we prove that power-law SULE yields geometric constraints on localization centers (such as their equidistribution) and quantitative bounds on eigenfunction correlators. As a dynamical consequence we obtain power-law localization in the sense of finite $q$-moments (up to a certain power $q$) of the position operator. Conversely, suitable bounds on eigenfunction correlators imply a corresponding form of power-law SULE, establishing a close connection between these notions. This highlights the role of power-law SULE as a structural mechanism governing localization beyond the exponential regime, including features typically associated with random operators, such as Anderson-type models. Our results reveal that power-law localization is intrinsically geometric: the spatial distribution of localization centers directly influences eigenfunction correlators and transport properties. As an application, we obtain power-law localization for long-range lattice operators with Stark-type potentials of sublinear growth whose spectral regime exhibits asymptotically collapsing spectral gaps and quasi-resonant structures, without relying on perturbative methods. Applications to long-range random operators are also discussed.

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

0 major / 3 minor

Summary. The paper develops a deterministic framework for lattice operators satisfying a power-law form of semi-uniform localization of eigenfunctions (SULE). It establishes that this assumption implies geometric constraints on the localization centers (including equidistribution), quantitative bounds on eigenfunction correlators, and dynamical localization in the form of finite q-moments of the position operator (up to a certain power q). A converse result is proved showing that suitable correlator bounds imply the corresponding power-law SULE. The framework is applied to long-range lattice operators with Stark-type potentials of sublinear growth that exhibit asymptotically collapsing spectral gaps, yielding power-law localization without perturbative methods; applications to long-range random operators are also discussed.

Significance. If the central derivations hold, the results provide a valuable deterministic bridge between spectral and dynamical aspects of power-law localization, emphasizing its geometric character through the distribution of centers. This extends structural insights beyond the exponential-decay regime and supplies non-perturbative consequences for models with quasi-resonant structures, which is relevant for both deterministic and random long-range operators. The explicit converse implication and the application to collapsing-gap Stark operators are notable strengths.

minor comments (3)
  1. The notation for the power-law exponent and the associated constants in the SULE definition (likely in §2) could be made more uniform across the statements of the main theorems to ease comparison between the direct and converse implications.
  2. Figure 1 (or the corresponding schematic of localization centers) would benefit from an explicit caption indicating the dependence on the power-law parameter; the current caption is brief and does not reference the relevant theorem.
  3. A short remark in the introduction or §1 comparing the obtained q-moment bound with the corresponding statement for exponential SULE would help situate the result for readers familiar with the classical literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment, including the recommendation to accept. We are pleased that the deterministic framework connecting power-law SULE to geometric constraints, correlator bounds, and dynamical localization (with the converse implication and the application to collapsing-gap Stark operators) was viewed as a valuable contribution.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation consists of deterministic implications starting from the external premise of power-law SULE (semi-uniform localization of eigenfunctions), from which geometric constraints on centers, quantitative bounds on correlators, and dynamical localization (finite q-moments) are derived. The converse direction is also established as a separate implication, but this is presented as a logical equivalence between related notions rather than a self-referential definition or fitted prediction. No equations or steps reduce by construction to the inputs, no self-citations are invoked as load-bearing uniqueness theorems, and the application to Stark-type operators is conditional on verifying the SULE hypothesis externally. The framework is self-contained against the stated assumptions without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no explicit free parameters, invented entities, or non-standard axioms are identifiable. The work extends the existing notion of SULE to a power-law regime using standard tools of spectral theory.

axioms (1)
  • standard math Standard results from functional analysis and spectral theory for self-adjoint operators on lattices
    The framework presupposes the usual Hilbert-space setting and spectral theorem for lattice operators.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.SP 2026-06 unverdicted novelty 6.0

    The paper proves asymptotic closeness of eigenvalues to the Stark ladder and power-law localization of eigenfunctions and the associated evolution for long-range discrete Dirac operators with Stark potential.

Reference graph

Works this paper leans on

38 extracted references · 6 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aizenman and S

    M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys.157(1993), 245–278. 17

    1993

  2. [2]

    Aizenman and S

    M. Aizenman and S. Warzel, Random Operators. Disorder Effects on Quantum Spectra and Dynamics, GSM 168, AMS (2015)

    2015

  3. [3]

    Aloisio, Dynamical localization and eigenvalue asymptotics: long-range hopping lattice operators with electric field

    M. Aloisio, Dynamical localization and eigenvalue asymptotics: long-range hopping lattice operators with electric field. Ann. Henri Poincar´ e (2026). https://doi.org/10.1007/s00023-026-01698-9

    work page doi:10.1007/s00023-026-01698-9 2026

  4. [4]

    Aloisio, S

    M. Aloisio, S. L. Carvalho, C. R. de Oliveira, Spectral Measures and Dynamics: Typical Behaviors. Berlin, Springer Nature (2023)

    2023

  5. [5]

    B. L. Altshuler and L. S. Levitov, Weak chaos in a quantum Kepler problem. Phys. Rep.288 (1997), 487–512

    1997

  6. [6]

    P. W. Anderson, Absence of diffusion in certain random lattices. Phys. Rev.109(1958), 1492–1505

    1958

  7. [7]

    Damanik, Schr¨ odinger operators with dynamically defined potentials, Ergodic Theory Dynam

    D. Damanik, Schr¨ odinger operators with dynamically defined potentials, Ergodic Theory Dynam. Systems.37(2017), 1681-1764

    2017

  8. [8]

    Damanik and J

    D. Damanik and J. Fillman, One-dimensional ergodic Schr¨ odinger operators, I. general theory, Graduate Studies in Mathematics221, American Mathematical Society, (2022)

    2022

  9. [9]

    Damanik and J

    D. Damanik and J. Fillman, One-dimensional ergodic Schr¨ odinger operators, II. specific classes, Graduate Studies in Mathematics249, American Mathematical Society, (2024)

    2024

  10. [10]

    F. A. B. F. de Moura and M. L. Lyra, Delocalization in the 1D Anderson model with long-range correlated disorder. Phys. Rev. Lett.81(1998), 3735

    1998

  11. [11]

    C. R. de Oliveira, Intermediate spectral theory and quantum dynamics. Progress in Math. Phys. Basel, Birkh¨ auser, (2009)

    2009

  12. [12]

    C. R. de Oliveira and M. Pigossi, Exact solvable family of discrete Schr¨ odinger operators with long-range hoppings. Math. Scand.130(2024), 527–537

    2024

  13. [13]

    C. R. de Oliveira and M. Pigossi, Point spectrum and SULE for time-periodic perturbations of discrete 1D Schr¨ odinger operators with electric fields. J. Stat. Phys.173(2018), 140–162

    2018

  14. [14]

    C. R. de Oliveira and M. Pigossi, Proof of dynamical localization for perturbations of discrete 1D Schr¨ odinger operators with uniform electric fields. Math. Z.291(2019), 1525–1541

    2019

  15. [15]

    Del Rio, S

    R. Del Rio, S. Jitomirskaya, Y. Last and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimension, rank one perturbations and localization. J. Anal. Math. 69(1996), 153–200

    1996

  16. [16]

    De Roeck, A

    W. De Roeck, A. Hannani, A. Lerose and N. Vandenbosch, Stark localization of interacting particles. Preprint. arXiv:2602.23352 [math-ph] (2026)

    work page arXiv 2026

  17. [17]

    Disertori, R

    M. Disertori, R. Maturana Escobar and C. Rojas-Molina, Decay of the Green’s function of the fractional Anderson model and connection to long-range SAW. J. Stat. Phys.191(2024), article number 33. 18

    2024

  18. [18]

    Fr¨ ohlich and T

    J. Fr¨ ohlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys.88(1983), 151–184

    1983

  19. [19]

    Gebert and C

    M. Gebert and C. Rojas-Molina, Lifshitz tails for the fractional Anderson model. J. Stat. Phys.179(2020), 341–353

    2020

  20. [20]

    Germinet, A

    F. Germinet, A. Kiselev and S. Tcheremchantsev, Transfer matrices and transport for Schr¨ odinger operators. Ann. Inst. Fourier.54(2004), 787–830

    2004

  21. [21]

    Germinet and A

    F. Germinet and A. Klein, Bootstrap multiscale analysis and localization in random media. (2001) Comm. Math. Phys.222, 415-448

    2001

  22. [22]

    Hu and Y

    S. Hu and Y. Sun, Localized state for nonlinear disordered stark model. Preprint. arXiv:2603.09243 [math.DS] (2026)

    work page arXiv 2026

  23. [23]

    Hu and Y

    S. Hu and Y. Sun, Wannier-Stark Localization for time quasi-periodic Hamiltonian operator onZ. Ann. Henri Poincar´ e (2025). https://doi.org/10.1007/s00023-024-01533-z

    work page doi:10.1007/s00023-024-01533-z 2025

  24. [24]

    Jian and Y

    J. Jian and Y. Sun, Dynamical localization for power-law long-range hopping random operators onZ 𝑑. Proc. Amer. Math. Soc.150(2022), 5369–5381

    2022

  25. [25]

    Jian and Y

    J. Jian and Y. Sun, Localization of interacting random particles with power-law long-range hopping. Ann. Henri Poincar´ e (2026). https://doi.org/10.1007/s00023-026-01726-8

    work page doi:10.1007/s00023-026-01726-8 2026

  26. [26]

    Kerner, O

    J. Kerner, O. Post, M. Sabri and M. Ta¨ ufer, The curious spectra and dynamics of non-locally finite crystals. Comm. Math. Phys.406(2025), article number 169

    2025

  27. [27]

    Von Dreifus and A

    H. Von Dreifus and A. Klein, A new proof of localization in the Anderson tight-binding model. 2(1989) Comm. Math. Phys. 124, 285-299

    1989

  28. [28]

    Kraisler, J

    J. Kraisler, J. Schenker and J. C. Schotland, Dynamic one-photon localization in a discrete model of quantum optics. SIAM J. Appl. Math.86(2026), no. 3, 776–790

    2026

  29. [29]

    H. N. Nazareno, C. A. A. da Silva, P. E. de Brito, Dynamical localization in aperiodic 1D systems under the action of electric fields. Superlattices Microstruct.18(1995), 297–307

    1995

  30. [30]

    Shi, A multi-scale analysis proof of the power-law localization for random operators onZ 𝑑

    Y. Shi, A multi-scale analysis proof of the power-law localization for random operators onZ 𝑑. J. Differ. Equ.297(2021), 201-225

    2021

  31. [31]

    Shi, Localization for almost-periodic operators with power-law long-range hopping: A Nash-Moser iteration type reducibility approach

    Y. Shi, Localization for almost-periodic operators with power-law long-range hopping: A Nash-Moser iteration type reducibility approach. Commun. Math. Phys.402(2023), 1765–1806

    2023

  32. [32]

    Shi and L

    Y. Shi and L. Wen, Diagonalization in a quantum kicked rotor model with non-analytic potential. J. Differ. Equ.355(2023), 334–368

    2023

  33. [33]

    Shi and L

    Y. Shi and L. Wen, Green’s function estimates for quasi-periodic operators onZ 𝑑 with power-law long-range hopping. Adv. Math.482(2025), 110661. 19

    2025

  34. [34]

    Shi and L

    Y. Shi and L. Wen, Localization for a class of discrete long-range quasi-periodic operators. Lett. Math. Phys.112(2022), Article number 86

    2022

  35. [35]

    Y. Shi, L. Wen and D. Yan, Localization for random operators onZ 𝑑 with the long-range hopping. Ann. Henri Poincar´ e (2025). https://doi.org/10.1007/s00023-025-01595-7

    work page doi:10.1007/s00023-025-01595-7 2025

  36. [36]

    Sun and C

    Y. Sun and C. Wang, Localization of polynomial long-range hopping lattice operator with uniform electric fields. Lett. Math. Phys.114(2024), Article number 6

    2024

  37. [37]

    Sun and C

    Y. Sun and C. Wang, Stark localization of Jacobi operator with applications to quantum spin models. Lett. Math. Phys.115(2025), Article number 126

    2025

  38. [38]

    Tcheremchantsev, How to prove dynamical localization

    S. Tcheremchantsev, How to prove dynamical localization. Commun. Math. Phys.221(2001), 27–56. Moacir Aloisio. Email: moacir.aloisio@ufvjm.edu.br, DME, UFVJM, Diamantina, MG, 39100-000, Brazil and ICMC, USP, S˜ ao Carlos, SP, 13566-590, Brazil C´ esar R. de Oliveira. Email: oliveira@ufscar.br, DM, UFSCar, S˜ ao Carlos, SP, 13560-970 Brazil Rodrigo Matos. E...

    2001