One-dimensional point interaction with three complex parameters

For a pair of non-Hermitian Hamiltonian H and its Hermitian adjoint H†, there are situations in which their eigenfunctions form a biorthogonal system. We illustrate such a situation by means of a one-particle system with a one-dimensional point interaction in the form of the Fermi pseudo-potential. The interaction consists of three terms with three strength parameters gi (i = 1, 2 and 3), which are all complex. This complex point interaction is neither Hermitian nor -invariant in general. The S-matrix for the transmission–reflection problem constructed with H (or with H†) in the usual manner is not unitary, but it conforms to the pseudo-unitarity that we define. The pseudo-unitarity is closely related to the biorthogonality of the eigenfunctions. The eigenvalue spectrum of H with the complex interaction is generally complex but there are cases where the spectrum is real. In such a case H and H† form a pseudo-Hermitian pair.