Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.2, pp.192--207

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2000, Volume 3, Number 2, pp.192--207

Schrödinger Difference Operators with Quadratic Potentials on Geometric Progressions.
Andreas Ruffing

We consider the spectral problem for a Schrödinger difference operator, being defined on stair functions on a geometric progression of type {+qⁿ, -qⁿ | n in Z }, q > 1. The potentials under consideration are quadratic and can be considered as discrete generalizations of the oscillator potential. In contrast to the corresponding situation in continuum quantum mechanics, the methods to determine the spectral properties turn out to be totally different. Existence results for eigensystems of the discrete oscillator are derived by combining results from the theory of bilateral Jacobi operators with efficient methods from the theory of holomorphic perturbations. The point spectrum of the Schrödinger operator under consideration turns out to grow exponentially. It can be used for regularizations in lattice quantum mechanics.

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