Numerical solution of Salpeter's equation

A method for solving Salpeter's relativistic bound-state equation is presented. The interaction can be given in either momentum space or configuration space and may have various Lorentz-Dirac properties. The operators of the equation are represented as matrices in a basis of nonrelativistic harmonic-oscillator states. The resulting non-Hermitian matrix is diagonalized for various values of the oscillator frequency, a variational parameter. To reduce the size of the matrices a two-particle Foldy-Wouthuysen transformation is applied. As an example, the charmonium and $b$-quarkonium mass spectra are calculated using a linear confining potential plus one-gluon exchange. The effects of the coupling between the positive- and negative-energy components are examined and found to be important for the light mesons.