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SIGMA, 2012, Volume 8, 061, 19 pages (Mi sigma738)  

This article is cited in 10 scientific papers (total in 10 papers)

Spectral analysis of certain Schrödinger operators

Mourad E.H. Ismaila, Erik Koelinkb

a Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
b Radboud Universiteit, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, the Netherlands

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].

Keywords: $J$-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.

DOI: https://doi.org/10.3842/SIGMA.2012.061

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Full text: http://emis.mi.ras.ru/.../061
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Bibliographic databases:

ArXiv: 1205.0821
MSC: 30E05; 33C45; 39A10; 42C05; 44A60
Received: May 7, 2012; in final form September 12, 2012; Published online September 15, 2012
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Citation: Mourad E.H. Ismail, Erik Koelink, “Spectral analysis of certain Schrödinger operators”, SIGMA, 8 (2012), 061, 19 pp.

Citation in format AMSBIB
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\by Mourad E.H. Ismail, Erik Koelink
\paper Spectral analysis of certain Schr\"{o}dinger operators
\jour SIGMA
\yr 2012
\vol 8
\papernumber 061
\totalpages 19
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Groenevelt W., Koelink E., “A Hypergeometric Function Transform and Matrix-Valued Orthogonal Polynomials”, Constr. Approx., 38:2 (2013), 277–309  crossref  mathscinet  zmath  isi  scopus
    2. Groenevelt W., Ismail M.E.H., Koelink E., “Spectral Decomposition and Matrix-Valued Orthogonal Polynomials”, Adv. Math., 244 (2013), 91–105  crossref  mathscinet  zmath  isi  scopus
    3. V. X. Genest, M. E. H. Ismail, L. Vinet, A. Zhedanov, “Tridiagonalization of the hypergeometric operator and the Racah–Wilson algebra”, Proc. Amer. Math. Soc., 144:10 (2016), 4441–4454  crossref  mathscinet  zmath  isi  elib  scopus
    4. F. A. Gruenbaum, L. Vinet, A. Zhedanov, “Tridiagonalization and the Heun equation”, J. Math. Phys., 58:3 (2017), 031703  crossref  mathscinet  zmath  isi  scopus
    5. S. Tsujimoto, L. Vinet, A. Zhedanov, “Tridiagonal representations of the q-oscillator algebra and Askey–Wilson polynomials”, J. Phys. A-Math. Theor., 50:23 (2017), 235202  crossref  mathscinet  zmath  isi  scopus
    6. Sh.-F. Tian, T.-T. Zhang, “Long-time asymptotic behavior for the Gerdjikov–Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition”, Proc. Amer. Math. Soc., 146:4 (2018), 1713–1729  crossref  mathscinet  zmath  isi  scopus
    7. X.-B. Wang, Sh.-F. Tian, T.-T. Zhang, “Characteristics of the breather and rogue waves in a (2+1)-dimensional nonlinear Schrödinger equation”, Proc. Amer. Math. Soc., 146:8 (2018), 3353–3365  crossref  mathscinet  zmath  isi  scopus
    8. Grunbaum F.A., Vinet L., Zhedanov A., “Algebraic Heun Operator and Band-Time Limiting”, Commun. Math. Phys., 364:3 (2018), 1041–1068  crossref  mathscinet  zmath  isi  scopus
    9. Alhaidari A.D., “Series Solutions of Heun-Type Equation in Terms of Orthogonal Polynomials”, J. Math. Phys., 59:11 (2018), 113507  crossref  mathscinet  zmath  isi  scopus
    10. Vinet L., Zhedanov A., “The Heun Operator of Hahn-Type”, Proc. Amer. Math. Soc., 147:7 (2019), 2987–2998  crossref  isi
  • Symmetry, Integrability and Geometry: Methods and Applications
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