Exact WKB Analysis of Eigenvalue Problems for an Ordinary Differential Equation Arising from the Mathematical Model of Mesons

A mass of meson, an interacting particle of mass intermediate between proton and neutron, is described by an eigenvalue problem of a certain second order ordinary differential equation. In this paper we consider this eigenvalue problem from the viewpoint of the exact WKB analysis. We find that a connection problem in the complex plane corresponds to this eigenvalue problem. Through studying this connection problem by using the exact WKB method, we give an explicit form of a secular equation for the eigenvalue problem modulo exponentially small terms. Comparisons with numerical calculations are also discussed.