Spectral analysis of non-commutative harmonic oscillators: the lowest eigenvalue and no crossing

TL;DR

This paper investigates the spectral properties of non-commutative harmonic oscillators, demonstrating the simplicity of the lowest eigenvalue and providing a matrix representation for numerical analysis.

Contribution

It introduces a decomposition of the oscillator into self-adjoint operators and establishes the simplicity of the lowest eigenvalue, with a numerical spectrum analysis.

Findings

The lowest eigenvalue is simple.

Decomposition into four self-adjoint operators.

Numerical spectrum analysis using Jacobi matrix.

Abstract

The lowest eigenvalue of non-commutative harmonic oscillators $Q$ is studied. It is shown that $Q$ can be decomposed into four self-adjoint operators, and all the eigenvalues of each operator are simple. We show that the lowest eigenvalue $E$ of $Q$ is simple. Furthermore a Jacobi matrix representation of $Q$ is given and spectrum of $Q$ is considered numerically.