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

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 Θneq 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 $Θ^{(Dirac}=I$) at the special, Hermitian-Hamiltonian-limit parameters.