Perturbation method for non-square Hamiltonians and its application to polynomial oscillators

TL;DR

The paper introduces an extended perturbation method for non-square Hamiltonians, particularly applicable to polynomial oscillators, involving a novel approach to eigenvalues as energy multiplets and a q-dimensional inversion process.

Contribution

It presents a new perturbation technique for non-square Hamiltonians with applications to polynomial oscillators, expanding the scope of traditional methods.

Findings

Extended perturbation method for non-square matrices.

Application to polynomial oscillator models.

Eigenvalues represented as energy multiplets.

Abstract

A remarkable extension of Rayleigh-Schroedinger perturbation method is found. Its (N+q) x (N+1) - dimensional Hamiltonians (as emerging, e.g., during quasi-exact constructions of bound states) are non-square matrices at q > 1. The role of an eigenvalue is played by an energy/coupling q-plet. In all orders, its perturbations are defined via a q-dimensional inversion.