The bound states of Dirac equation with a scalar potential

We study the bound states of the 1+1 dimensional Dirac equation with a scalar potential, which can also be interpreted as a position dependent "mass'', analytically as well as numerically. We derive a Prüfer-like representation for the Dirac equation, which can be used to derive a condition for the existence of bound states in terms of the fixed point of the nonlinear Prüfer equation for the angle variable. Another condition was derived by interpreting the Dirac equation as a Hamiltonian flow on the 2-dimensional Euclidean space and a shooting argument for the induced flow on the space of its Lagrangian planes following a similar calculation by Jones (Ergodic Theor Dyn Syst, 8 (1988) 119-138). The two conditions are shown to be equivalent, and used to compute the bound states analytically and numerically, as well as to derive a Calogero-like upper bound on the number of bound states. The analytic computations are also compared to the bound states computed using techniques from supersymmetric quantum mechanics