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 $H\ne H^{†}$ with the real and explicit spectrum given by the zeros of the $N-$th Chebyshev polynomial of the first kind. The second item $\Theta \ne I$ 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 $H^{†}\Theta =\Theta H$. 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, $\vec{\kappa }\in {\mathcal{D}}^{(physical)}$.

16 pp., 4 figs

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