Spectral zeta functions of a 1D Schrödinger problem

Details Spectral zeta functions of a 1D Schrödinger problem Spectral zeta functions of a 1D Schrödinger problem

2011-10-10

arxiv.org/abs/1110.2004v1

Physics

Mathematical Physics

13 pages

Scientific paper

We study the spectral zeta functions associated to the radial Schr\"odinger problem with potential $V(x)=x^{2M}+\alpha x^{M-1} +(\lambda^2-1/4)/x^2$. 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 ${}_5F_4$ hypergeometric series as an example. Our work is then extended to a class of related ${\cal PT}$-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 $G_n$ which appear in the associated integrable quantum field theory.

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