Counting operator analysis of the discrete spectrum of some model Hamiltonians

TL;DR

This paper presents a method for analyzing the spectrum of model Hamiltonians using counting operators and commutator relations, leading to a decomposition into multiplets and the concept of stable eigenstates.

Contribution

It introduces a systematic approach to spectrum analysis via counting operators, commutator conditions, and stable eigenstates, which is a novel framework for understanding Hamiltonian spectra.

Findings

Spectrum decomposed into multiplets based on reference Hamiltonian properties

Stable eigenstates can be used to generate new eigenstates under weak conditions

Method applicable to a class of model Hamiltonians with specific commutator relations

Abstract

The first step in the counting operator analysis of the spectrum of any model Hamiltonian H is the choice of a Hermitean operator M in such a way that the third commutator with H is proportional to the first commutator. Next one calculates operators R and R^\dagger which share some of the properties of creation and annihilation operators, and such that $M$ becomes a counting operator. The spectrum of H is then decomposed into multiplets, not determined by the symmetries of H, but by those of a reference Hamiltonian H_ref, which is defined by H_ref=H-R-R^\dagger, and which commutes with M. Finally, we introduce the notion of stable eigenstates. It is shown that under rather weak conditions one stable eigenstate can be used to construct another one.