A Generalized Family of Discrete PT-symmetric Square Wells

Abstract

N-site-lattice Hamiltonians H(N) are introduced and perceived as a set of systematic discrete approximants of a certain PT-symmetric square-well-potential model with the real spectrum and with a non-Hermiticity which is localized near the boundaries of the interval. Its strength is controlled by one, two or three parameters. The problem of the explicit construction of a nontrivial metric which makes the theory unitary is then addressed. It is proposed and demonstrated that due to the not too complicated (viz., tridiagonal matrix) form of our input Hamiltonians, the computation of the metric is straightforward and that its matrix elements prove obtainable, non-numerically, in elementary polynomial forms. © 2013 Springer Science+Business Media New York.