Stability of Salpeter Solutions

In the framework of ‘instantaneous approximations’ to the Bethe–Salpeter formalism for the description of bound states within quantum field theories, depending on the Lorentz structure of the Bethe–Salpeter interaction kernel solutions of the (full) Salpeter equation with some confining interactions may exhibit certain instabilities [1], possibly related to the Klein paradox, signalling the decay of states assumed to be bound by these confining interactions, and observed in numerical (variational) studies [1] of the Salpeter equation. The (presumably) simplest scenario allowing for the fully analytic investigation of this problem is set by the reduced Salpeter equation [2] with harmonic-oscillator interaction. In this case, the integral equation of Salpeter simplifies to either an algebraic relation or a second-order homogeneous linear ordinary differential equation, immediately accessible to standard techniques. There one can hope to be able to decide unambiguously whether this setting poses a well-defined (eigenvalue) problem the solutions of which correspond to stable bound states associated to real energy eigenvalues that are bounded from below.