Delta-function potential with a complex coupling

Abstract

We explore the Hamiltonian operator H = - d(2)/dx(2) + z delta(x), where x is an element of R, delta(x) is the Dirac delta function and z is an arbitrary complex coupling constant. For a purely imaginary z, H has a spectral singularity at E = - z(2)/4 is an element of R+. For Re( z) < 0, H has an eigenvalue at E = - z(2)/4. For the case that Re(z) > 0, H has a real, positive, continuous spectrum that is free from spectral singularities. For this latter case, we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive-definite inner product that renders H self-adjoint. This allows us to address the intriguing question of the nonlocal aspects of the equivalent Hermitian Hamiltonian for the system. In particular, we compute the energy expectation values for various Gaussian wave packets to show that the non-Hermiticity effect diminishes rapidly outside an effective interaction region.