Asymptotics of the eigenvalues of the rotating harmonic oscillator

Abstract

The eigenenergies λ of a radial Schrödinger equation associated with the problem of a rotating harmonic oscillator are studied, these being values which admit eigensolutions which vanish at both the origin (a regular singularity of the equation) and at infinity. Asymptotic expansions, for the case where a coupling parameter α is small, are derived for λ. The approximation for λ consists of two components, an asymptotic expansion in powers of α, and a single term which is exponentially small (which can be associated with tunneling effects). The method of proof is rigorous, and utilizes three separate asymptotic approximations for the eigenfunction in the complex radial plane, involving elementary functions (WKB or Liouville-Green approximations), a modified Bessel function and a parabolic cylinder function.