Spectral zeta functions of a 1D Schr\"odinger problem
Abstract
We study the spectral zeta functions associated to the radial Schr\"odinger problem with potential V(x)=x2M+alpha xM-1+(lambda2-1/4)/x2. Using the quantum Wronskian equation, we provide results such as closed-form evaluations for some of the second zeta functions i.e. the sum over the inverse eigenvalues squared. Also we discuss how our results can be used to derive relationships and identities involving special functions, using a particular 5F4 hypergeometric series as an example. Our work is then extended to a class of related PT-symmetric eigenvalue problems. Using the fused quantum Wronskian we give a simple method for calculating the related spectral zeta functions. This method has a number of applications including the use of the ODE/IM correspondence to compute the (vacuum) nonlocal integrals of motion Gn which appear in an associated integrable quantum field theory.