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arXiv · 2209.04608

Eigenvalue Fluctuations of 1-dimensional random Schrödinger operators

Takuto Mashiko·Yuma Marui·Naoki Maruyama·Fumihiko Nakano
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Abstract

As an extension to the paper by Breuer, Grinshpon, and White \cite{B}, 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 \cite{B} 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 $α\le α_c$, where $α_c$ differs depending on $f$. Furthermore we study the asymptotic behavior of its expectation value.

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Takuto Mashiko, Yuma Marui, Naoki Maruyama, Fumihiko Nakano. 2022-09-10. Eigenvalue Fluctuations of 1-dimensional random Schrödinger operators. https://arxiv.org/abs/2209.04608

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