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        Difference Schr\"odinger Operators\\[0.3cm]
        for Harmonic Oscillators on a Unitary Lattice\\[0.9cm]
       
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        ANDREAS L. RUFFING\\[0.9cm]                        
        {\em Munich University of Technology}\\
        {\em Department of Mathematics}\\
        {\em Boltzmannstra{\ss}e 3}\\
        {\em D-85747 Garching, Germany}\\
        {\em ruffing@ma.tum.de}\\
        {\em http://www-m6.ma.tum.de/$\sim$ruffing/}\\[0.7cm]
       
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The formalism of raising and lowering operators is developed for the difference operator analogue of a quantum harmonic oscillator which acts on functions on a discrete support. The grid under consideration is a mixed version of an equidistant lattice and a basic linear grid. Several properties of the grid are described. The grids under consideration are referred to by the name unitary linear lattices. The ladder difference operators are derived and compared with the continuum situation. The arising spectral problems for these operators are dealt by using the theory of bilateral Jacobi operators in weighted $l^{2}({\Bbb{Z}})$ spaces. Eventual applications to mathematical physics and numerical Schr\"odinger theory are briefly discussed.\\[0.1cm]


{\bf References and Literature for Further Reading}\\



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\itm{[1]} {R. N. {\'A}lvarez, D. Bonatsos and Yu. F. Smirnov, $q$-Deformed vibron model for diatomic molecules, {\em Physical Review A.} 50 (1994) 1088-1095.}\\
\itm{[2]} {R. Askey, S. K. Suslov, The $q$-harmonic oscillator and the Al-Salam and Carlitz polynomials, {\em Letters in Mathematical Physics} 29 (1993) No. 2, 123-132.}\\
\itm{[3]} {C. Berg, M.E.H. Ismail, Q-Hermite polynomials and classical orthogonal polynomials, {\em Can. J. Math.} 48 (1996) 43-63.}\\
\itm{[4]} {C. Berg, A. Ruffing, Generalized $q$-Hermite Polynomials, {\em Communications in Mathematical Physics} {\bf 223} (2001) 1, 29-46.}\\
\itm{[5]} {A. Ruffing, J. Lorenz, K. Ziegler, Difference Ladder Operators for a Harmonic Schr\"odinger Oscillator Using Unitary Linear Lattices, {\em Journal of Computational and Applied Mathematics} 153 (2003), 395-410.}

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