General pseudo self-adjoint boundary conditions for a 1D KFG particle in a box

We consider a 1D Klein–Fock–Gordon particle in a finite interval, or box. We construct for the first time the most general set of pseudo self-adjoint boundary conditions for the Hamiltonian operator that is present in the first order in time 1D Klein–Fock–Gordon wave equation, or the 1D Feshbach–Villars wave equation. We show that this set depends on four real parameters and can be written in terms of the one-component wavefunction for the second order in time 1D Klein–Fock–Gordon wave equation and its spatial derivative, both evaluated at the endpoints of the box. Certainly, we write the general set of pseudo self-adjoint boundary conditions also in terms of the two-component wavefunction for the 1D Feshbach–Villars wave equation and its spatial derivative, evaluated at the ends of the box; however, the set actually depends on these two column vectors each multiplied by the singular matrix that is present in the kinetic energy term of the Hamiltonian. As a consequence, we found that the two-component wavefunction for the 1D Feshbach–Villars equation and its spatial derivative do not necessarily satisfy the same boundary condition that these quantities satisfy when multiplied by the singular matrix. In any case, given a particular boundary condition for the one-component wavefunction of the standard 1D Klein–Fock–Gordon equation and using the pair of relations that arise from the very definition of the two-component wavefunction for the 1D Feshbach–Villars equation, the respective boundary condition for the latter wavefunction and its derivative can be obtained. Our results can be extended to the problem of a 1D Klein–Fock–Gordon particle moving on a real line with a point interaction (or a hole) at one point.