  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.
