Asymptotics of determinants of discrete Schrödinger operators

Abstract

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

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

We calculate $\lim \det T_n(f;\epsilon )/G(f{)}^n$ and show that this limit can be adjusted to any positive number by shifting $\epsilon$, even though the asymptotic eigenvalue distribution of $T_n(f;\epsilon )$ does not depend on $\epsilon$. Secondly, we derive a formula for the asymptotics of $\det T_n(f)/G(f{)}^n$ when $f$ has jump discontinuities. In this case the asymptotics depend on the fractional part of $cn$, where $c$ is a point of discontinuity.