Discrete quantum square well of the first kind

A toy-model quantum system is proposed. At a given integer N it is defined by the pair of N by N real matrices (H,Θ) of which the first item H specifies an elementary, diagonalizable non-Hermitian Hamiltonian H neq H^dagger with the real and explicit spectrum given by the zeros of the N-th Chebyshev polynomial of the first kind. The second item Θneq I must be (and is being) constructed as the related Hilbert-space metric which specifies the (in general, non-unique) physical inner product and which renders our toy-model Hamiltonian selfadjoint, i.e., compatible with the Dieudonne equation H^dagger Θ= Θ H. The elements of the (in principle, complete) set of the eligible metrics are then constructed in closed band-matrix form. They vary with our choice of the N-plet of optional parameters, Θ=Θ\(vecκ\)>0 which must be (and are being) selected as lying in the positivity domain of the metric, vecκ in {cal D}^(physical).