Asymptotics of determinants of discrete Schrödinger operators

  • Alain Bourget

    California State University, Fullerton, USA
  • Tyler McMillen

    California State University, Fullerton, USA
Asymptotics of determinants of discrete Schrödinger operators cover
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Abstract

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

in terms of the values of , where 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 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