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

http://arxiv.org/abs/1105.1863
Quantum Physics (quant-ph); High Energy Physics – Lattice (hep-lat); Mathematical Physics (math-ph)