Exactly Solvable Many-Body Systems and Pseudo-Hermitian Point Interactions

The complex generalization of conventional quantum mechanics has been investigated extensively in recent years. In particular it is shown that the standard formulation of quantum mechanics in terms of Hermitian Hamiltonians is overly restrictive and a consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry) [1]. It is proven that if PT symmetry is not spontaneously broken, the dynamics of a non-Hermitian Hamiltonian system is still governed by unitary time evolution. A number of models with PT-symmetric and continuous interaction potentials have been constructed and studied [2]. In this article we study Hamiltonian systems with singular interaction potentials at a point. We give a systematic and complete description of the boundary conditions and the spectra properties for self-adjoint, PT-symmetric systems and systems with real spectra. We then study the integrability of one dimensional many body systems with these kinds of point interactions.