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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[1] and Tyler McMillen[2]

(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 [3] 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