Solvability and PT-symmetry in a double-well model with point interactions

We show that and how point interactions offer one of the most suitable guides towards a quantitative analysis of properties of certain specific non-Hermitian (usually called PT-symmetric) quantum-mechanical systems. A double-well model is chosen, an easy solvability of which clarifies the mechanisms of the unavoided level crossing and of the spontaneous PT-symmetry breaking. The latter phenomenon takes place at a certain natural boundary of the domain of the "acceptable" parameters of the model. Within this domain the model mediates a nice and compact explicit illustration of the not entirely standard probabilistic interpretation of the physical bound states in the very recently developed (so called PT symmetric or, in an alternative terminology, pseudo-Hermitian) new, fairly exciting and very quickly developing branch of Quantum Mechanics.Comment: 24 p., written for the special journal issue "Singular Interactions in Quantum Mechanics: Solvable Models". Will be also presented to the int. conference "Pseudo-Hermitian Hamiltonians in Quantum Physics III" (Instanbul, Koc University, June 20 - 22, 2005) http://home.ku.edu.tr/~amostafazadeh/workshop/workshop.ht