Spectral functions for the Schrödinger operator on R+ with a singular potential

In this article we analyze the spectral zeta function, the heat kernel, and the resolvent of the operator −d2/dr2+κ/r2+r2 over the interval (0,∞) for κ≥−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∊N0, do occur. This implies that the small-t asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as tk/(ln t)ℓ+1 for k,ℓ∊N0. The corresponding statements for the resolvent are also shown. Furthermore, we evaluate the zeta determinant of the operator for all values of κ and any self-adjoint extension.