Spectral properties of the massless relativistic harmonic oscillator

The spectral properties of the pseudo-differential operator(−d2/dx2)1/2+x2(−d2/dx2)1/2+x2 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.