Exact solution of the 1D Dirac equation for a pseudoscalar interaction potential with the inverse-square-root variation law.

We present the exact solution of the one-dimensional stationary Dirac equation for the pseudoscalar interaction potential, which consists of a constant and a term that varies in accordance with the inverse-square-root law. The general solution of the problem is written in terms of irreducible linear combinations of two Kummer confluent hypergeometric functions and two Hermite functions with non-integer indices. Depending on the value of the indicated constant, the effective potential for the Schrödinger-type equation to which the problem is reduced can form a barrier or well. This well can support an infinite number of bound states. We derive the exact equation for the energy spectrum and construct a rather accurate approximation for the energies of bound states. The Maslov index involved turns out to be non-trivial; it depends on the parameters of the potential.