Spectral singularities, biorthonormal systems and a two-parameter family of complex point interactions

Abstract

A curious feature of complex scattering potentials nu(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., - d(2)/dx(2) + nu(x), and its adjoint. We establish the equivalence of this description with the mathematicians' definition of spectral singularities for the potential nu(x) = z_delta(x+a)+ z(+)delta(x-a), where z(+/-) and a are respectively complex and real parameters and delta(x) is the Dirac delta function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z(+/-) where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing PT-symmetry.