Discrete Schrödinger operators with decaying and oscillating potentials

Frank, R.; Larson, S.

doi:10.1090/spmj/1803

Discrete Schrödinger operators with decaying and oscillating potentials
HTML articles powered by AMS MathViewer

by R. L. Frank and S. Larson

St. Petersburg Math. J. 35 (2024), 233-244

DOI: https://doi.org/10.1090/spmj/1803

Published electronically: April 12, 2024

HTML | PDF | Request permission

Abstract:

The paper is devoted to a family of discrete one-dimensional Schrödinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential $V(n)=\lambda n^{-\alpha }\cos (\pi \omega n^\beta )$ with $1<\beta <2\alpha$, it is proved that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.

References

Michael Christ and Alexander Kiselev, Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: some optimal results, J. Amer. Math. Soc. 11 (1998), no. 4, 771–797. MR 1621882, DOI 10.1090/S0894-0347-98-00276-8
P. Deift and R. Killip, On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. Math. Phys. 203 (1999), no. 2, 341–347. MR 1697600, DOI 10.1007/s002200050615
François Delyon, Hervé Kunz, and Bernard Souillard, One-dimensional wave equations in disordered media, J. Phys. A 16 (1983), no. 1, 25–42. MR 700179, DOI 10.1088/0305-4470/16/1/012
F. Delyon, B. Simon, and B. Souillard, From power-localized to extended states in a class of one-dimensional disordered systems, Phys. Rev. Lett. 52 (1984), no. 24, 2187–2189.
François Delyon, Barry Simon, and Bernard Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré Phys. Théor. 42 (1985), no. 3, 283–309 (English, with French summary). MR 797277
Michael S. P. Eastham and Hubert Kalf, Schrödinger-type operators with continuous spectra, Research Notes in Mathematics, vol. 65, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667015
Rupert L. Frank and Simon Larson, On the spectrum of the Kronig-Penney model in a constant electric field, Probab. Math. Phys. 3 (2022), no. 2, 431–490. MR 4452513, DOI 10.2140/pmp.2022.3.431
Rupert L. Frank and Barry Simon, Eigenvalue bounds for Schrödinger operators with complex potentials. II, J. Spectr. Theory 7 (2017), no. 3, 633–658. MR 3713021, DOI 10.4171/JST/173
D. J. Gilbert and D. B. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), no. 1, 30–56. MR 915965, DOI 10.1016/0022-247X(87)90212-5
Thierry Jecko, On Schrödinger and Dirac operators with an oscillating potential, Rev. Roumaine Math. Pures Appl. 64 (2019), no. 2-3, 283–314. With contributions by Aiman Mbarek. MR 4012606
A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Comm. Math. Phys. 179 (1996), no. 2, 377–400. MR 1400745, DOI 10.1007/BF02102594
Alexander Kiselev, Stability of the absolutely continuous spectrum of the Schrödinger equation under slowly decaying perturbations and a.e. convergence of integral operators, Duke Math. J. 94 (1998), no. 3, 619–646. MR 1639550, DOI 10.1215/S0012-7094-98-09425-X
Alexander Kiselev, Yoram Last, and Barry Simon, Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Comm. Math. Phys. 194 (1998), no. 1, 1–45. MR 1628290, DOI 10.1007/s002200050346
S. Kotani and N. Ushiroya, One-dimensional Schrödinger operators with random decaying potentials, Comm. Math. Phys. 115 (1988), no. 2, 247–266. MR 931664, DOI 10.1007/BF01466772
Helge Krüger, Schrödinger operators with potential $V(n)=n^{-\gamma }\cos (2\pi n^\rho )$, Entropy and the quantum II, Contemp. Math., vol. 552, Amer. Math. Soc., Providence, RI, 2011, pp. 109–116. MR 2868043, DOI 10.1090/conm/552/10912
Milivoje Lukic, Orthogonal polynomials with recursion coefficients of generalized bounded variation, Comm. Math. Phys. 306 (2011), no. 2, 485–509. MR 2824479, DOI 10.1007/s00220-011-1287-9
S. N. Naboko, On the dense point spectrum of Schrödinger and Dirac operators, Teoret. Mat. Fiz. 68 (1986), no. 1, 18–28 (Russian, with English summary). MR 875178
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 493421
Christian Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Comm. Math. Phys. 193 (1998), no. 1, 151–170. MR 1620313, DOI 10.1007/s002200050322
Barry Simon, Some Jacobi matrices with decaying potential and dense point spectrum, Comm. Math. Phys. 87 (1982/83), no. 2, 253–258. MR 684102, DOI 10.1007/BF01218563
Barry Simon, Some Schrödinger operators with dense point spectrum, Proc. Amer. Math. Soc. 125 (1997), no. 1, 203–208. MR 1346989, DOI 10.1090/S0002-9939-97-03559-4
Günter Stolz, Spectral theory for slowly oscillating potentials. I. Jacobi matrices, Manuscripta Math. 84 (1994), no. 3-4, 245–260. MR 1291120, DOI 10.1007/BF02567456
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
J. von Neumann and E. P. Wigner, Über merkwürdige diskrete Eigenwerte, Phys. Z. 30 (1929), 465–467.
Denis A. W. White, Schrödinger operators with rapidly oscillating central potentials, Trans. Amer. Math. Soc. 275 (1983), no. 2, 641–677. MR 682723, DOI 10.1090/S0002-9947-1983-0682723-7

AMS Website Down

St. Petersburg Mathematical Journal

Discrete Schrödinger operators with decaying and oscillating potentials
HTML articles powered by AMS MathViewer

Abstract: