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Spectral functions for the Schrödinger operator on \(\mathbb{R}^+\) with a singular potential. (English) Zbl 1310.34117

Summary: In this article we analyze the spectral zeta function, the heat kernel, and the resolvent of the operator \(-d^2/dr^2 + \kappa/r^2+r^2\) over the interval \((0,\infty)\) for \(\kappa \geq -1/4\). Depending on the self-adjoint extension chosen, nonstandard properties of the zeta function and of asymptotic properties of the heat kernel and resolvent are observed. In particular, for the zeta function nonstandard locations of poles as well as logarithmic branch cuts at \(s = -k,\; k \in \mathbb{N}_0\), do occur. This implies that the small-\(t\) asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as \(t^k/(\ln t)^{\ell+1}\; \text{for}\; k,\ell \in \mathbb{N}_0\). The corresponding statements for the resolvent are also shown. Furthermore, we evaluate the zeta determinant of the operator for all values of \(\kappa\) and any self-adjoint extension.{
©2010 American Institute of Physics}

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
35K08 Heat kernel
37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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