Quasi-exact solution of the problem of relativistic bound states in the $(1{+}1)$-dimensional case

We investigate the problem of bound states for bosons and fermions in the framework of the relativistic configurational representation with the kinetic part of the Hamiltonian containing purely imaginary finite shift operators $e^{\pm i\hbar d/dx}$ instead of differential operators. For local $($quasi$)$potentials of the type of a rectangular potential well in the $(1{+}1$)-dimensional case, we elaborate effective methods for solving the problem analytically that allow finding the spectrum and investigating the properties of wave functions in a wide parameter range. We show that the properties of these relativistic bound states differ essentially from those of the corresponding solutions of the Schrödinger and Dirac equations in a static external potential of the same form in a number of fundamental aspects both at the level of wave functions and of the energy spectrum structure. In particular, competition between $\hbar$ and the potential parameters arises, as a result of which these distinctions are retained at low-lying levels in a sufficiently deep potential well for $\hbar\ll1$ and the boson and fermion energy spectra become identical.