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 $(H,\mathrm{\Theta })$ of which the first item $H$ specifies an elementary, diagonalizable non-Hermitian Hamiltonian $H \neq H^\dagger$withtherealandexplicitspectrumgivenbythezerosofthe$N-$th Chebyshev polynomial of the first kind. The second item $\mathrm{\Theta }\mathrm{\ne }I$ 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 $H^{†}\mathrm{\Theta }=\mathrm{\Theta }H$. 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, $\mathrm{\Theta }=\mathrm{\Theta }(\vec{\kappa })>0$ which must be (and are being) selected as lying in the positivity domain of the metric, $\vec{\kappa} \in {\cal D}^{(physical)}$.