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

Abstract

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 $N=2,3,\ldots$, 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.