As an extension to the paper by Breuer et al., Ann. Henri Poincare 22, 3763 (2021), we study the linear statistics for the eigenvalues of the Schrödinger operator with random decaying potential with order (α > 0) at infinity. We first prove similar statements as in Breuer et al., Ann. Henri Poincare 22, 3763 (2021) for the trace of f(H), where f belongs to a class of analytic functions: there exists a critical exponent αc such that the fluctuation of the trace of f(H) converges in probability for α > αc, and satisfies a central limit theorem statement for α ≤ αc, where αc differs depending on f. Furthermore we study the asymptotic behavior of its expectation value.
REFERENCES
1.
Billingsley
, P.
, Probability and Measure
, 3rd ed., Wiley Series in Probability and Mathematical Statistics
(A Wiley-Interscience Publisher
, 1995
).2.
Breuer
, J.
, Grinshpon
, Y.
, and White
, M. J.
, “Spectral fluctuations for Schrödinger operators with a random decaying potential
,” Ann. Henri Poincare
22
, 3763
–3794
(2021
).3.
Delyon
, F.
, Simon
, B.
, and Souillard
, B.
, “From power pure point to continuous spectrum in disordered systems
,” Ann. Henri Poincare
42
, 283
–309
(1985
).4.
Dumitriu
, I.
and Edelman
, A.
, “Matrix models for beta ensembles
,” J. Math. Phys.
43
, 5830
–5847
(2002
).5.
Durrett
, R.
, Probability: Theory and Examples
, 5th ed., Cambridge Series in Statistical and Probabilistic Mathematics
(Cambridge University Press
, 2019
).6.
Kiselev
, A.
, Last
, Y.
, and Simon
, B.
, “Modified prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators
,” Commun. Math. Phys.
194
, 1
–45
(1998
).7.
Kotani
, S.
and Nakano
, F.
, “Level statistics for the one-dimensional Schroedinger operators with random decaying potential
,” Interdiscip. Math. Sci.
17
, 343
–373
(2014
).8.
Kotani
, S.
and Nakano
, F.
, “Poisson statistics for 1d Schrödinger operators with random decaying potentials
,” Electron. J. Probab.
22
, 1
–31
(2017
).9.
Kritchevski
, E.
, Valkó
, B.
, and Virág
, B.
, “The scaling limit of the critical one-dimensional random Schrödinger operator
,” Commun. Math. Phys.
314
, 775
–806
(2012
).10.
Nakano
, F.
, “Level statistics for one-dimensional Schrödinger operators and Gaussian beta ensemble
,” J. Stat. Phys.
156
, 66
–93
(2014
).11.
Nakano
, F.
, “Fluctuation of density of states for 1d Schrödinger operators
,” J. Stat. Phys.
166
, 1393
–1404
(2017
).12.
Nakano
, F.
, “Shape of eigenvectors for the decaying potential model
,” Ann. Henri Poincaré.
24
, 871
–893
(2023
); arXiv:2203.03125.13.
Rifkind
, B.
and Virág
, B.
, “Eigenvectors of the 1-dimensional critical random Schrödinger operator
,” Geom. Funct. Anal.
28
, 1394
–1419
(2018
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
