PT-symmetric operators and metastable states of the 1D relativistic oscillators

Abstract

We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second-order separated operators. Such operators, defined by complex dilation, give the resonances of the problem. Their unique extensions as closed operators with a purely point spectrum are PT-symmetric with positive eigenvalues converging to the Schrödinger ones as c → ∞. Precise numerical computations show that these eigenvalues coincide with the positions of the resonances up to the order of the width. The corresponding eigenfunctions are a definite choice of metastable states of the problem. Similar results are found for the Klein–Gordon oscillator: here also we have two closed, isospectral and complex conjugate extensions of the formal operator with PT-symmetry, but an infinite number of self-adjoint extensions and physical dynamics. The infinitely many pairs of eigenvectors of the two closed PT-symmetric operators give metastable states for any choice of the dynamics. The eigenvalues of the operator defined by complex dilation are resonances, although not according to the standard definition, for any dynamics.