Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 061, 19 pages      arXiv:1205.0821      https://doi.org/10.3842/SIGMA.2012.061

Spectral Analysis of Certain Schrödinger Operators

Mourad E.H. Ismail a and Erik Koelink b
a) Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
b) Radboud Universiteit, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, the Netherlands

Received May 07, 2012, in final form September 12, 2012; Published online September 15, 2012

Abstract
The J-matrix method is extended to difference and q-difference operators and is applied to several explicit differential, difference, q-difference and second order Askey-Wilson type operators. The spectrum and the spectral measures are discussed in each case and the corresponding eigenfunction expansion is written down explicitly in most cases. In some cases we encounter new orthogonal polynomials with explicit three term recurrence relations where nothing is known about their explicit representations or orthogonality measures. Each model we analyze is a discrete quantum mechanical model in the sense of Odake and Sasaki [J. Phys. A: Math. Theor. 44 (2011), 353001, 47 pages].

Key words: J-matrix method; discrete quantum mechanics; diagonalization; tridiagonalization; Laguere polynomials; Meixner polynomials; ultraspherical polynomials; continuous dual Hahn polynomials; ultraspherical (Gegenbauer) polynomials; Al-Salam-Chihara polynomials; birth and death process polynomials; shape invariance; zeros.

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