Discrete quantum square well of the first kind

Miloslav Znojil

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, \(\vec{\kappa} \in {\cal D}^{(physical)}\).

Quantum Physics (quant-ph); High Energy Physics – Lattice (hep-lat); Mathematical Physics (math-ph)

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