Asymptotics of determinants of discrete Schrödinger operators
Alain Bourget
California State University, Fullerton, USATyler McMillen
California State University, Fullerton, USA
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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 , where is a constant. We extend this result in two ways: First, we consider shifting the index: Let
We calculate and show that this limit can be adjusted to any positive number by shifting , even though the asymptotic eigenvalue distribution of does not depend on . Secondly, we derive a formula for the asymptotics of when has jump discontinuities. In this case the asymptotics depend on the fractional part of , where is a point of discontinuity.
Cite this article
Alain Bourget, Tyler McMillen, Asymptotics of determinants of discrete Schrödinger operators. J. Spectr. Theory 8 (2018), no. 4, pp. 1617–1634
DOI 10.4171/JST/237