Spectral Properties of the Massless Relativistic Harmonic Oscillator

The spectral properties of the pseudo-differential operator $(-d^2\mathrm{/}d{x}^2{)}^{1\mathrm{/}2}+x^2$ 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.