Spectral Properties of the Massless Relativistic Harmonic Oscillator

TL;DR

This paper investigates the spectral characteristics of a massless relativistic harmonic oscillator operator, providing eigenvalue representations, asymptotic formulas, and heat kernel estimates using advanced analytical techniques.

Contribution

It introduces new analytical methods to characterize eigenvalues and eigenfunctions of the operator, including asymptotic and trace formulas.

Findings

Eigenvalues and eigenfunctions explicitly represented.

Precise asymptotic formulas derived.

Trace asymptotics and heat kernel estimates established.

Abstract

The spectral properties of the pseudo-differential operator $(-d^2/dx^2)^{1/2}+x^2$ are analyzed by a combination of functional integration methods and direct analysis. We obtain a representation of its eigenvalues and eigenfunctions, prove precise asymptotic formulae, and establish various analytic properties. We also derive trace asymptotics and heat kernel estimates.