Spectral Analysis of Certain Schrödinger Operators

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].

References in corpus (4)

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Cited by in corpus (8)

• Algebraic Heun operator and band-time limiting
• Tridiagonalization and the Heun equation
• Series solutions of Heun-type equation in terms of orthogonal polynomials
• Series solution of a ten-parameter second order differential equation with three regular and one irregular singularities
• Spectral decomposition and matrix-valued orthogonal polynomials
• On the zeros of some families of polynomials satisfying a three-term recurrence associated to Gribov operator
• Progressive approximation of bound states by finite series of square-integrable functions
• The Kontorovich-Lebedev transform as a map between $d$-orthogonal polynomials