Asymptotics of determinants of discrete Schr\"odinger operators

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.