## Solvable non-Hermitian discrete square well with closed-form physical inner product

Miloslav Znojil

A non-Hermitian N−level quantum model with two free real parameters is proposed in which the bound-state energies are given as roots of an elementary trigonometric expression and in which they are, in a physical domain of parameters, all real. The wave function components are expressed as closed-form superpositions of two Chebyshev polynomials. In any eligible physical Hilbert space of finite dimension $$N<\infty$$ our model is constructed as unitary with respect to an underlying Hilbert-space metric $$\Theta\neq I$$. The simplest version of the latter metric is finally constructed, at any dimension N=2,3,…, in closed form. This version of the model may be perceived as an exactly solvable N−site lattice analogue of the $$N=\infty$$ square well with complex Robin-type boundary conditions. At any $$N<\infty$$ our closed-form metric becomes trivial (i.e., equal to the most common Dirac’s metric $$\Theta(Dirac)=I$$) at the special, Hermitian-Hamiltonian-limit parameters.

http://arxiv.org/abs/1409.3788
Mathematical Physics (math-ph); Quantum Physics (quant-ph)