Discrete quantum square well of the first kind

Abstract: A toy-model quantum system is proposed. At a given integer $N$ it is defined by the pair of $N$ by $N$ real matrices $(H,\Theta)$ 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 $\Theta\neq I$ 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 $H^\dagger \Theta= \Theta\,H$. 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, $\Theta=\Theta(\vec{\kappa})>0$ which must be (and are being) selected as lying in the positivity domain of the metric, $\vec{\kappa} \in {\cal D}^{(physical)}$.