# Asymptotics of determinants of discrete Schrödinger operators

### Alain Bourget

California State University, Fullerton, USA### Tyler McMillen

California State University, Fullerton, USA

## Abstract

We consider the asymptotics of the determinants of large discrete Schrödinger operators, i.e. "discrete Laplacian $+$ diagonal":

\[ T_n(f) = -[\delta_{j,j+1}+\delta_{j+1,j}] + \mbox{diag}(f(1/n), f(2/n),\dots, f(n/n)) \]We extend a result of M. Kac [3] who found a formula for

in terms of the values of $f$, where $G(f)$ is a constant. We extend this result in two ways: First, we consider shifting the index: Let

\[ T_n(f;\varepsilon) = -[\delta_{j,j+1}+\delta_{j+1,j}] + \mbox{diag}\Big(f\Big(\frac{\varepsilon}{n}\Big), f\Big(\frac{1+ \varepsilon}{n}\Big), \dots , f\Big(\frac{n-1+ \varepsilon}{n}\Big)\Big). \]We calculate $limdetT_{n}(f;ε)/G(f)_{n}$ and show that this limit can be adjusted to any positive number by shifting $ε$, even though the asymptotic eigenvalue distribution of $T_{n}(f;ε)$ does not depend on $ε$. Secondly, we derive a formula for the asymptotics of $detT_{n}(f)/G(f)_{n}$ when $f$ has jump discontinuities. In this case the asymptotics depend on the fractional part of $cn$, where $c$ is a point of discontinuity.

## Cite this article

Alain Bourget, Tyler McMillen, Asymptotics of determinants of discrete Schrödinger operators. J. Spectr. Theory 8 (2018), no. 4, pp. 1617–1634

DOI 10.4171/JST/237