Asymptotics of determinants of discrete Schr\"odinger operators

TL;DR

This paper extends classical results on the asymptotics of determinants of large discrete Schrödinger operators by considering index shifts and jump discontinuities in the potential function, revealing new dependence on these modifications.

Contribution

It generalizes Kac's formula for determinants by analyzing shifted operators and operators with discontinuous potentials, showing how these affect asymptotic behavior.

Findings

The limit of the determinant ratio can be any positive number through index shifting.

Asymptotics depend on the fractional part of the product of the discontinuity point and n.

The eigenvalue distribution remains unaffected by the index shift.

Abstract

We consider the asymptotics of the determinants of large discrete Schr\"odinger operators, i.e. "discrete Laplacian $+$ diagonal": \[T_n(f) = -[\delta_{j,j+1}+\delta_{j+1,j}] + \mbox{diag}\left(f\left(\frac{1}{n}\right), f\left(\frac{2}{n}\right),\dots, f\left(\frac{n}{n}\right)\right) \] We extend a result of M. Kac, who found a formula for \[\lim_{n\rightarrow\infty} \frac{\det(T_n(f))}{G(f)^n} \] 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 \[T_n(f;\varepsilon) = -[\delta_{j,j+1}+\delta_{j+1,j}] + \mbox{diag}\left(f\left(\frac{\varepsilon}{n}\right), f\left(\frac{1+ \varepsilon}{n}\right), \dots, f\left(\frac{n-1+ \varepsilon}{n}\right)\right) \] We calculate $\lim \det T_n(f;\varepsilon)/G(f)^n$ and show that this limit can be any positive number by shifting $\varepsilon$, even though the asymptotic eigenvalue distribution of $T_n(f;\varepsilon)$ does not depend on $\varepsilon$. 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 $c n$, where $c$ is a point of discontinuity.