Eigenvalue Fluctuations of 1dimensional random Schrödinger operators
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 ${\cal O}(x^{\alpha})$ ($\alpha>0$) 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 $\alpha_c$ such that the fluctuation of the trace of $f(H)$ converges in probability for $\alpha > \alpha_c$, and satisfies a CLT statement for $\alpha \le \alpha_c$, where $\alpha_c$ differs depending on $f$. Furthermore we study the asymptotic behavior of its expectation value.
 Publication:

arXiv eprints
 Pub Date:
 September 2022
 DOI:
 10.48550/arXiv.2209.04608
 arXiv:
 arXiv:2209.04608
 Bibcode:
 2022arXiv220904608M
 Keywords:

 Mathematical Physics;
 Mathematics  Probability