Exact solution of the E⊗ε Jahn-Teller and Rabi Hamiltonian by generalised spheroidal wavefunctions?

Abstract

The Schrodinger equation for the E⊗ε Jahn-Teller and Rabi systems in Bargmann's Hilbert space (1962) is a system of two ordinary differential equations of first order for the spin up and down components of the wavefunctions. This system has two regular and one irregular singular points. The energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions. The differential equations of the generalised spheroidal wavefunctions have the same singular points and the same exponents at each singular point. It is therefore conjectured that the component wavefunctions in the excited state i can be expanded in i+4 generalised spheroidal wavefunctions. The energy eigenvalues ν⁽ⁱ⁾ (i=0,1, . . ., 5) calculated with the conjectured component wavefunctions agree with numerical values within the computational error. The same is true for the coefficients of the Neumann expansion of the component wavefunctions. A proof is still missing.