Padé oscillators and a new formulation of perturbation theory

Abstract

Being guided by the problem of bound states in potentials close to their Padé approximants, a new Rayleigh–Schrödinger-type perturbation theory is developed. The unperturbed system is understood: here in a broader sense: its solutions are not needed, but merely the related nondiagonal unperturbed propagator R. In particular, all the chain models H0ψ=ES0ψ (H0, S0=band matrices) with arbitrary perturbations are then perturbatively solvable, with R constructed in terms of auxiliary matrix continued fraction fn. Alternatively, a ‘‘generalized unperturbed spectrum’’ f̂n may be required as an input: The algebraically constructed asymptotics of the fn’s play this role in our Padé examples. Due to S≠I, the‘‘Sturmians’’ may also be constructed. In the test evaluations of the binding energies and/or couplings, the simultaneous upper and lower bounds of high precision are shown to be numerically obtainable.