Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Dispersive Decay Estimates for periodic Jacobi operators on the half-line

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Periodic Jacobi operators on the half-line satisfy t^{-1/2} weighted dispersive decay in full generality and t^{-1/3} or t^{-1/(q+1)} global decay under explicit spectral conditions.

desk verdict The local t^{-1/2} decay is solid and new; the global t^{-1/3} proof has a repairable uniformity gap. read the letter →

arxiv 2505.14498 v1 pith:6I2TARAB submitted 2025-05-20 math.SP math-phmath.CAmath.MP

Amir Sagiv , Remy Kassem , Michael I Weinstein This is my paper · ORCID
classification math.SPmath-phmath.CAmath.MP
keywords decayoperatorsdispersiveestimateshalf-lineinftyjacobiperiodic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Imagine a wave traveling on a very long chain of sites, where each site is connected to its nearest neighbors. The chain has a repeating pattern of connection strengths and on-site energies, so it is described by a periodic Jacobi operator. The paper studies how solutions of the linear Schrodinger equation on such a chain spread as time grows. The evolution is unitary, so the total size of the wave never changes, but spreading lowers the height of its peaks.

The chain has an endpoint, and that breaks translation symmetry. The standard Fourier transform used on doubly infinite lines is therefore not available. The authors instead write the solution using orthogonal polynomials naturally associated with the Jacobi operator, expressing the time evolution as an integral over the spectrum. Periodicity makes the spectral measure and the phase functions explicit, and classical oscillatory integral estimates convert those integrals into decay rates.

The main results are three decay rates. For initial data localized near the endpoint, the wave peak decays at least like time^{-1/2} for every period. For the unweighted global maximum, the rate is time^{-1/3} under a condition on the phase function, or time^{-1/(q+1)} when the period q is even and the spectrum is exactly q separated bands. These are the first dispersive decay bounds for such half-line periodic Jacobi problems.

Extended reading notes

Core claim

Theorem 3.1: for every periodic Jacobi operator J of the form (1.1), the local weighted estimate ∥e^{-itJ}P_c u∥_{ℓ∞_{-1}} ≤ C t^{-1/2}∥u∥_{ℓ1_1} holds. If the paper is correct, this estimate holds for all periods, including closed-gap cases, because every bounded spectrum has at least one band edge where stationary phase gives t^{-1/2}.

Load-bearing premise

The oscillatory reduction rests on imported spectral facts in Theorem 2.1, especially the density formula dμ_c(x)=√(4-Δ^2)/|t_{2,1}(x)| dx and the nonvanishing, sign-constant behavior of t_{2,1} on each band, cited to [43, Thm. 10.76]. If t_{2,1} had an interior zero, the amplitude Q(y)=Δ'(y)L(y)/t_{2,1}(y) in (4.9) would be singular and the Van der Corput bounds in Section 5 would not follow. For the global t^{-1/3} result, the explicit nondegeneracy condition k''_j=0 implies k'''_j≠0 is also load-bearing, and its genericity is left open in the paper.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies dispersive decay for e^{-itJ}P_c where J is a periodic Jacobi operator on ℓ²(N). Using the explicit eigenfunction expansion of J in terms of transfer matrices and the discriminant Δ, the authors reduce the propagator to finite sums of oscillatory integrals with phase -t k_j(φ)+ℓφ. Theorem 3.1 claims a weighted ℓ¹₁→ℓ^∞_{-1} bound t^{-1/2} for every period q. Theorem 3.2 claims a global ℓ¹→ℓ^∞ bound t^{-1/3} under the nondegeneracy condition 'if k_j''=0 then k_j'''≠0', and t^{-1/(q+1)} for even q when the spectrum has exactly q disjoint bands. The paper is self-contained after importing standard spectral facts on periodic Jacobi operators.

Significance. If valid, Theorem 3.1 appears to be the first dispersive estimate of this type for periodic Jacobi operators on the half-line and gives a uniform t^{-1/2} local decay independent of the band structure; the global results connect to known rates for periodic Schrödinger operators on Z and to edge-mode asymptotics. The proof strategy is transparent and parameter-free: it reduces the problem to controlled oscillatory integrals and explicitly identifies the roles of gapped versus ungapped band edges. The paper also honestly states open questions, e.g., genericity of the nondegeneracy condition and the correct rate when it fails, and it cites parallel work [14]. However, as detailed below, the written derivations of the simplified amplitude and of the uniformity in Theorem 3.2(1) need correction.

major comments (3)
  1. [Section 4.1, Eqs. (4.7)–(4.9)] The change of variables leading to the simplified propagator is not correct as written. Since √(4−Δ²)=2|sinΘ| with Θ∈[−π,0], the density (2.10) is proportional to (−sinΘ)/t_{2,1} dx, not to 1/(t_{2,1} sinΘ) dx as printed in (4.7). Moreover, from (4.10), k′_j(φ)=−2 sinφ/Δ′_j(k_j(φ)), so dx=−2 sinφ/Δ′_j(k_j(φ)) dφ, not Δ′/(−2 sinφ)dφ as stated before (4.8). With these corrections one obtains an amplitude proportional to L/(t_{2,1}Δ′) rather than Q=Δ′L/t_{2,1} in (4.8)–(4.9). Because Q is used in every subsequent Van der Corput estimate, this algebraic discrepancy must be resolved; as it stands the derivation of (4.8) does not follow from (4.6).
  2. [Section 5.2, Eqs. (5.15)–(5.17)] The proof of Theorem 3.2(1) contains a uniformity gap. After (5.15), the phase in (5.16) is λ(φ)=−t k_j(φ)+ℓφ. The text states that for all t≠t_i there is a lower bound on |∂_φ λ| and hence Lemma 5.2 gives a t^{-1} rate; this is false uniformly in ℓ and t, because for any t one can choose ℓ∈Z with t k′_j(φ_p)−ℓ arbitrarily small. The gap is repairable: the cutoff η in (5.12) keeps the support of X_j away from T_3, so |k_j'''|≥c>0 there, and since λ'''=−t k_j''', Lemma 5.1 with s=3 applies directly to each term in (5.16) for every t and every ℓ. The theorem is plausible, but the proof as written should be replaced by this direct application.
  3. [Section 5.2, proof of (3.2)] The final paragraph of the proof of Theorem 3.2(2) is too compressed to verify the claimed t^{-1/(q+1)} rate. It says that a point where only a nonzero lower bound on k^{(q)} is available yields t^{-1/(q+1)}, but Lemma 5.1 with s=q gives t^{-1/q}; also the partition of [−π,0] according to which derivative is bounded below is not constructed, and the uniformity of the constants and cutoffs is not shown. Please expand this proof or clarify the exponent.
minor comments (5)
  1. [Section 4.1, after Eq. (4.6)] The phrase 'assume without loss of generality that t1,2 > 0' should refer to t_{2,1}, the entry that appears in the density formula (2.10); this is a typo in the index.
  2. [Section 5.2, Eq. (5.12)] The definition of η uses a minimum over T_2, but the case T_2=∅ is not handled; if k_j'' has no zeros then F^{(s)}_0 is identically zero and the argument is trivial, but this case should be stated explicitly.
  3. [Lemma 4.7, proof] The line 'k(ℓ)_j(φ0)=0 for for some 2≤ℓ≤q' should read 'for every 2≤ℓ≤q', and the duplicated 'for' should be removed.
  4. [Theorem 2.1(2)] The phrase 'J has only continuous spectrum and pure point spectrum' is imprecise; it should say 'absolutely continuous spectrum and pure point spectrum', since the continuous spectrum is shown to be purely absolutely continuous.
  5. [Remark 3.3] The claimed 'more elaborate and concrete computation of the propagator' for the SSH model is not identified; a pointer to the relevant equations or a short derivation would help the reader verify the t^{-1/3} claim.
Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central estimates carry no fitted constants. They rest on standard spectral theory of Jacobi operators, one explicit nondegeneracy hypothesis in Theorem 3.2(1), and classical oscillatory integral lemmas. No new entities or parameters are introduced.

assumptions (6)
  • standard math Every bounded Jacobi operator on ℓ2(N) has δ1 as a cyclic vector and admits the generalized eigenfunction expansion (2.5).
    Invoked in Proposition 4.1 to represent e^{-itJ}P_c as an integral against the spectral measure; the proof is cited to [7] rather than given.
  • domain assumption For a q-periodic Jacobi operator, σ_c(J) is the union of q bands, Δ(x)=Tr T_q(x) has degree q, Θ_j is a monotone reparametrization of each band, and dμ_c(x)=√(4-Δ^2)/|t_{2,1}(x)| dx.
    Imported from Theorem 2.1, citing [43,56]; this band structure and density are the starting point of the oscillatory integral reduction in Section 4.
  • domain assumption t_{2,1}(x) does not vanish and does not change sign inside any band I_j.
    Used after Eq. (4.6) to drop the absolute value in (2.10) and define the smooth amplitude Q in (4.9); cited to [43, Thm. 10.76].
  • domain assumption If σ_c(J) is a single interval, J is a discrete Schrödinger operator with constant coefficients (Flaschka-Borg).
    Used in Section 3 before Theorem 3.1 to ensure a gapped band edge exists for minimal period q≥2, making the t^{-1/2} rate dominate.
  • domain assumption For Theorem 3.2(1), at every point where k''_j(φ)=0, one has k'''_j(φ)≠0.
    Explicit hypothesis of the theorem; the paper states that whether it is always or generically true remains open.
  • standard math Van der Corput's lemma and the nonstationary-phase integration-by-parts bound hold for the phases and amplitudes that arise.
    These are the engine of Section 5; the paper states them as Lemmas 5.1 and 5.2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dispersive Decay Estimates for periodic Jacobi operators on the half-line." pith.science (2026). https://pith.science/paper/6I2TARAB

@misc{pith2026250514498,
  author       = {Pith},
  title        = {Pith review of: Dispersive Decay Estimates for periodic Jacobi operators on the half-line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6I2TARAB}},
  note         = {Machine review of arXiv:2505.14498}
}
abstract

We establish dispersive time-decay estimates for periodic Jacobi operators on the discrete half-line, $\N$. Specifically, we prove $t^{-1/2}$ decay in the weighted $\ell^\infty_{-1}$ norm for all such operators. For the global $\ell^1 \to \ell^\infty$ decay estimate, we show that $t^{-1/3}$ decay holds under a nondegeneracy condition on the discriminant. Alternatively, for any even period $q\geq2$, if the continuous spectrum consists of exactly $q$ disjoint intervals (bands), we obtain a $t^{-1/(q+1)}$ decay rate without any further assumptions.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Sharp Polynomial Velocity Decay Bounds for Multidimensional Periodic Schr\"odinger Operators

    math-ph 2025-09 conditional novelty 7.0 of 10

    The group velocity and Lieb-Robinson velocity of a periodic Schrodinger operator decay as mu^{-p0+1} in the large-coupling limit, where p0 is the minimal period.

Reference graph

Works this paper leans on

64 extracted references · 63 canonical work pages · cited by 1 Pith paper

  1. [14]

    H. L. Cycon and B. Simon. Schr¨ odinger operators: with applications to quantum mechanics and global geometry . Springer Science & Business Media, 1987

    work page 1987

  2. [1]

    Slow propagation velocities in Schr¨ odinger operators with large periodic potential

    H. Abdul-Rahman et al. “Slow propagation velocities in Schr¨ odinger operators with large periodic potential”. Annales Henri Poincar´ e(2024), pp. 1–29

    work page 2024

  3. [2]

    M. J. Ablowitz and H. Segur. Solitons and the inverse scattering transform . SIAM, 1981

    work page 1981

  4. [3]

    Amrein and V

    W. Amrein and V. Georgescu. Characterization of bound states and scattering states in quan- tum mechanics. Tech. rep. Univ., Geneva, 1973. REFERENCES 15

    work page 1973

  5. [4]

    J. K. Asb´ oth, L. Oroszl´ any, and A. P´ alyi.A Short Course on Topological Insulators . Vol. 919

    work page

  6. [5]

    Lecture Notes in Physics, Springer, 2016

    work page 2016

  7. [6]

    Dispersive estimate for quasi-periodic Schr¨ odinger operators on 1-d lattices

    D. Bambusi and Z. Zhao. “Dispersive estimate for quasi-periodic Schr¨ odinger operators on 1-d lattices”. Advances in Mathematics 366 (2020), p. 107071

    work page 2020

  8. [7]

    Non-diffracting states in one-dimensional Floquet photonic topological insu- lators

    M. Bellec et al. “Non-diffracting states in one-dimensional Floquet photonic topological insu- lators”. Europhysics Letters 119.1 (2017), p. 14003

    work page 2017

Show all 64 references
  1. [8]

    I. M. Berezanskii. Expansions in eigenfunctions of selfadjoint operators . Vol. 17. American Mathematical Society, 1968

    1968

  2. [9]

    Complete ionization for a non-autonomous point interaction model in d= 2

    W. Borrelli, R. Carlone, and L. Tentarelli. “Complete ionization for a non-autonomous point interaction model in d= 2”. Communications in Mathematical Physics 395.2 (2022), pp. 963– 1005

    2022

  3. [10]

    Topological Classification of Insulators: I. Non-interacting Spectrally-Gapped One-Dimensional Systems

    J.-H. Chung and J. Shapiro. “Topological Classification of Insulators: I. Non-interacting Spectrally-Gapped One-Dimensional Systems”. arXiv preprint arXiv:2306.00268 (2023)

    2023 arXiv

  4. [11]

    Ionization in a 1-dimensional dipole model

    O. Costin, J. Lebowitz, and C. Stucchio. “Ionization in a 1-dimensional dipole model”. Reviews in Mathematical Physics 20.07 (2008), pp. 835–872

    2008

  5. [12]

    Nonperturbative time dependent solution of a simple ionization model

    O. Costin, R. D. Costin, and J. L. Lebowitz. “Nonperturbative time dependent solution of a simple ionization model”. Communications in Mathematical Physics 361 (2018), pp. 217–238

    2018

  6. [13]

    Resonance theory for Schr¨ odinger operators

    O. Costin and A. Soffer. “Resonance theory for Schr¨ odinger operators”. Communications in Mathematical Physics 224 (2001), pp. 133–152

    2001

  7. [15]

    Damanik, J

    D. Damanik, J. Fillman, and G. Young. Optimal Dispersion for Discrete Periodic Schr¨ odinger Operators. preprint. 2025

    2025

  8. [16]

    Quantum dynamics of periodic and limit-periodic Jacobi and block Jacobi matrices with applications to some quantum many body problems

    D. Damanik, M. Lukic, and W. Yessen. “Quantum dynamics of periodic and limit-periodic Jacobi and block Jacobi matrices with applications to some quantum many body problems”. Communications in Mathematical Physics 337.3 (2015), pp. 1535–1561

    2015

  9. [17]

    What is ballistic transport?

    D. Damanik, T. Malinovitch, and G. Young. “What is ballistic transport?” Journal of Spectral Theory (2024)

    2024

  10. [18]

    Properties of the scattering matrix and dispersion estimates for Jacobi operators

    I. Egorova, M. Holzleitner, and G. Teschl. “Properties of the scattering matrix and dispersion estimates for Jacobi operators”. Journal of Mathematical Analysis and Applications 434.1 (2016), pp. 956–966

    2016

  11. [19]

    Dispersion estimates for one-dimensional discrete Schr¨ odinger and wave equations

    I. Egorova, E. A. Kopylova, and G. Teschl. “Dispersion estimates for one-dimensional discrete Schr¨ odinger and wave equations”.Journal of Spectral Theory 5.4 (2015), pp. 663–696

    2015

  12. [20]

    Asymptotic completeness for quantum mechanical potential scattering: I. Short range potentials

    V. Enss. “Asymptotic completeness for quantum mechanical potential scattering: I. Short range potentials”. Communications in Mathematical Physics 61.3 (1978), pp. 285–291

    1978

  13. [21]

    Dispersive estimates for massive Dirac opera- tors in dimension two

    M. B. Erdogan, W. R. Green, and E. Toprak. “Dispersive estimates for massive Dirac opera- tors in dimension two”. Journal of Differential Equations 264.9 (2018), pp. 5802–5837

    2018

  14. [22]

    The massless Dirac equation in two dimen- sions: zero-energy obstructions and dispersive estimates

    M. B. Erdo˘ gan, M. Goldberg, and W. R. Green. “The massless Dirac equation in two dimen- sions: zero-energy obstructions and dispersive estimates”. Journal of Spectral Theory 11.3 (2021), pp. 935–979

    2021

  15. [23]

    Dispersive estimates for massive Dirac opera- tors in dimension two

    M. B. Erdo˘ gan, W. R. Green, and E. Toprak. “Dispersive estimates for massive Dirac opera- tors in dimension two”. Journal of Differential Equations 264.9 (2018), pp. 5802–5837

    2018

  16. [24]

    Dispersive estimates for Dirac operators in dimension three with obstructions at threshold energies

    M. B. Erdo˘ gan, W. R. Green, and E. Toprak. “Dispersive estimates for Dirac operators in dimension three with obstructions at threshold energies”. American Journal of Mathematics 141.5 (2019), pp. 1217–1258. 16 REFERENCES

    2019

  17. [25]

    Ballistic Transport for Periodic Jacobi Operators on

    J. Fillman. “Ballistic Transport for Periodic Jacobi Operators on”. In: From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory: A Volume in Honor of Lance Littlejohn ’s 70th Birthday. Springer, 2021, pp. 57–68

    2021

  18. [26]

    Discrete and periodic illustrations of some aspects of the inverse method

    H. Flaschka. “Discrete and periodic illustrations of some aspects of the inverse method”. Dy- namical Systems, Theory and Applications: Battelle Seattle 1974 Rencontres (2005), pp. 441– 466

    2005

  19. [27]

    Twisted equivariant matter

    D. S. Freed and G. W. Moore. “Twisted equivariant matter”. In: Annales Henri Poincar´ e. Vol. 14. 8. Springer. 2013, pp. 1927–2023

    2013

  20. [28]

    Calculation of Gauss quadrature rules

    G. H. Golub and J. H. Welsch. “Calculation of Gauss quadrature rules”. Mathematics of computation 23.106 (1969), pp. 221–230

    1969

  21. [29]

    B. C. Hall. Quantum theory for mathematicians . Springer, 2013

    2013

  22. [30]

    Radiative decay of edge states in Floquet media

    S. N. Hameedi, A. Sagiv, and M. I. Weinstein. “Radiative decay of edge states in Floquet media”. Multiscale Modeling & Simulation 21.3 (2023), pp. 925–963

    2023

  23. [31]

    Resolvent expansions for the Schr¨ odinger operator on the discrete half-line

    K. Ito and A. Jensen. “Resolvent expansions for the Schr¨ odinger operator on the discrete half-line”. Journal of Mathematical Physics 58.5 (2017)

    2017

  24. [32]

    Spectral properties of Schr¨ odinger operators and time-decay of the wave functions

    A. Jensen and T. Kato. “Spectral properties of Schr¨ odinger operators and time-decay of the wave functions”. Duke Math. J. 46.1 (1979), pp. 583–611

    1979

  25. [33]

    Quantized nonlinear Thouless pumping

    M. J¨ urgensen, S. Mukherjee, and M. C. Rechtsman. “Quantized nonlinear Thouless pumping”. Nature 596.7870 (2021), pp. 63–67

    2021

  26. [34]

    Time decay estimates for discrete semi-infinite dimer (SSH) Hamiltonians

    R. Kassem, A. Sagiv, and M. I. Weinstein. “Time decay estimates for discrete semi-infinite dimer (SSH) Hamiltonians”. In preparation. 2025

    2025

  27. [35]

    Metastable states in parametrically excited multimode Hamilton- ian systems

    E. Kirr and M. Weinstein. “Metastable states in parametrically excited multimode Hamilton- ian systems”. Communications in mathematical physics 236 (2003), pp. 335–372

    2003

  28. [36]

    Weighted energy decay for 1D Klein–Gordon equation

    A. Komech and E. Kopylova. “Weighted energy decay for 1D Klein–Gordon equation”. Com- munications in Partial Differential Equations 35.2 (2010), pp. 353–374

    2010

  29. [37]

    Dispersive estimates for 1D discrete Schr¨ odinger and Klein–Gordon equations

    A. I. Komech, E. A. Kopylova, and M. Kunze. “Dispersive estimates for 1D discrete Schr¨ odinger and Klein–Gordon equations”. Applicable Analysis 85.12 (2006), pp. 1487–1508

    2006

  30. [38]

    Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator

    T. Koornwinder, A. Kostenko, and G. Teschl. “Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator”. Advances in Mathematics 333 (2018), pp. 796–821

    2018

  31. [39]

    Kopylova and A

    E. Kopylova and A. Komech. Dispersion Decay and Scattering Theory . Wiley, 2014

    2014

  32. [40]

    Marchenko-Ostrovski mappings for periodic Jacobi matri- ces

    E. Korotyaev and A. Kutsenko. “Marchenko-Ostrovski mappings for periodic Jacobi matri- ces”. Russian Journal of Mathematical Physics 14 (2007), pp. 448–452

    2007

  33. [41]

    Dispersion estimates for the discrete Laguerre operator

    A. Kostenko and G. Teschl. “Dispersion estimates for the discrete Laguerre operator”. Letters in Mathematical Physics 106 (2016), pp. 545–555

    2016

  34. [42]

    Dispersive decay estimates for Dirac equations with a domain wall

    J. Kraisler, A. Sagiv, and M. I. Weinstein. “Dispersive decay estimates for Dirac equations with a domain wall”. SIAM Journal on Mathematical Analysis 56.6 (2024), pp. 7194–7227

    2024

  35. [43]

    On the Time-decay of solutions arising from periodically forced Dirac Hamiltonians

    J. Kraisler, A. Sagiv, and M. I. Weinstein. “On the Time-decay of solutions arising from periodically forced Dirac Hamiltonians”. arXiv preprint arXiv:2501.07466. To appear in J Diff Eq (2025)

    2025 arXiv

  36. [44]

    M. Luki´ c. A First Course in Spectral Theory. Vol. 226. American Mathematical Society, 2022

    2022

  37. [45]

    Observation of the topological soliton state in the Su–Schrieffer–Heeger model

    E. J. Meier, F. A. An, and B. Gadway. “Observation of the topological soliton state in the Su–Schrieffer–Heeger model”. Nature communications 7.1 (2016), p. 13986

    2016

  38. [46]

    Observation of Floquet states in graphene

    M. Merboldt et al. “Observation of Floquet states in graphene”. Nature Physics (2025), pp. 1– 7. REFERENCES 17

    2025

  39. [47]

    Dispersive estimates for periodic discrete one-dimensional Schr¨ odinger operators

    Y. Mi and Z. Zhao. “Dispersive estimates for periodic discrete one-dimensional Schr¨ odinger operators”. Proceedings of the American Mathematical Society 150.1 (2022), pp. 267–277

    2022

  40. [48]

    Metastability of breather modes of time- dependent potentials

    P. D. Miller, A. Soffer, and M. I. Weinstein. “Metastability of breather modes of time- dependent potentials”. Nonlinearity 13.3 (2000), p. 507

    2000

  41. [49]

    P. D. Miller. Applied asymptotic analysis . Vol. 75. American Mathematical Soc., 2006

    2006

  42. [50]

    Fast algorithms using orthogonal polynomials

    S. Olver, R. M. Slevinsky, and A. Townsend. “Fast algorithms using orthogonal polynomials”. Acta Numerica 29 (2020), pp. 573–699

    2020

  43. [51]

    Edge-mode lasing in 1D topological active arrays

    M. Parto et al. “Edge-mode lasing in 1D topological active arrays”. Physical review letters 120.11 (2018), p. 113901

    2018

  44. [52]

    On the spectral theory and dispersive estimates for a dis- crete Schr¨ odinger equation in one dimension

    D. E. Pelinovsky and A. Stefanov. “On the spectral theory and dispersive estimates for a dis- crete Schr¨ odinger equation in one dimension”.Journal of mathematical physics 49.11 (2008)

    2008

  45. [53]

    A remark on bound states in potential-scattering theory

    D. Ruelle. “A remark on bound states in potential-scattering theory”. Il Nuovo Cimento A (1965-1970) 61.4 (1969), pp. 655–662

    1969

  46. [54]

    Effective gaps in continuous Floquet Hamiltonians

    A. Sagiv and M. I. Weinstein. “Effective gaps in continuous Floquet Hamiltonians”. SIAM Journal on Mathematical Analysis 54.1 (2022), pp. 986–1021

    2022

  47. [55]

    Dispersive estimates for Schr¨ odinger operators: a survey

    W. Schlag. “Dispersive estimates for Schr¨ odinger operators: a survey”. Mathematical aspects of nonlinear dispersive equations 163 (2007), pp. 255–285

    2007

  48. [56]

    Is the continuum SSH model topological?

    J. Shapiro and M. I. Weinstein. “Is the continuum SSH model topological?” Journal of Math- ematical Physics 63.11 (2022)

    2022

  49. [57]

    B. Simon. Szeg˝ o’s Theorem and its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials. Princeton university press, 2010

    2010

  50. [58]

    Resonance and Radiation damping

    A. Soffer and M. Weinstein. “Resonance and Radiation damping...” Inventiones Math. (1999)

    1999

  51. [59]

    E. M. Stein and T. S. Murphy. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Vol. 3. Princeton University Press, 1993

    1993

  52. [60]

    Solitons in polyacetylene

    W.-P. Su, J. R. Schrieffer, and A. J. Heeger. “Solitons in polyacetylene”. Physical review letters 42.25 (1979), p. 1698

    1979

  53. [61]

    G. Teschl. Jacobi operators and completely integrable nonlinear lattices . 72. American Math- ematical Soc., 2000

    2000

  54. [62]

    M. Toda. Theory of nonlinear lattices . Vol. 20. Springer Science & Business Media, 2012

    2012

  55. [63]

    The race to compute high-order Gauss–Legendre quadrature

    A. Townsend. “The race to compute high-order Gauss–Legendre quadrature”. SIAM News 48.2 (2015), pp. 1–3

    2015

  56. [64]

    Topological acoustics

    H. Xue, Y. Yang, and B. Zhang. “Topological acoustics”. Nature Reviews Materials 7.12 (2022), pp. 974–990. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA Email address: amir.sagiv@njit.edu Department of Applied Physics and Applie...

    2022

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.