Eigenvalue Fluctuations of 1-dimensional random Schrödinger operators

2022/09/10 by Takuto Mashiko, Mashiko, Takuto, Yuma Marui +5
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum chaos and dynamical systems #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2209.04608

openalex publication_date 2022/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

As an extension to the paper by Breuer, Grinshpon, and White \citeB, we study the linear statistics for the eigenvalues of the Schrödinger operator with random decaying potential with order \cal O(x^-α) (α>0) at infinity. We first prove similar statements as in \citeB for the trace of f(H), where f belongs to a class of analytic functions : there exists a critical exponent α_c such that the fluctuation of the trace of f(H) converges in probability for α> α_c, and satisfies a CLT statement for α≤ α_c, where α_c differs depending on f. Furthermore we study the asymptotic behavior of its expectation value.

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