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# Journal of Spectral Theory

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Volume 8, Issue 4, 2018, pp. 1617–1634
DOI: 10.4171/JST/237

Published online: 2018-10-22

Asymptotics of determinants of discrete Schrödinger operators

Alain Bourget and Tyler McMillen

(1) California State University, Fullerton, USA
(2) California State University, Fullerton, USA

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  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 $T_n(f;\varepsilon) = -[\delta_{j,j+1}+\delta_{j+1,j}] + \mbox{diag}\Big(f\Big(\frac{\varepsilon}{n}\Big), f\Big(\frac{1+ \varepsilon}{n}\Big), \dots , f\Big(\frac{n-1+ \varepsilon}{n}\Big)\Big).$ We calculate $\lim \det T_n(f;\varepsilon)/G(f)^n$ and show that this limit can be adjusted to 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.

Keywords: Schrödinger operators, determinants, Szegő’s limit theorem

Bourget Alain, McMillen Tyler: Asymptotics of determinants of discrete Schrödinger operators. J. Spectr. Theory 8 (2018), 1617-1634. doi: 10.4171/JST/237