RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 SIGMA: Year: Volume: Issue: Page: Find

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

Spectral analysis of certain Schrödinger operators

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

Full text: PDF file (450 kB)
Full text: http://emis.mi.ras.ru/.../061
References: PDF file   HTML file

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
Language:

Citation: Mourad E.H. Ismail, Erik Koelink, “Spectral analysis of certain Schrödinger operators”, SIGMA, 8 (2012), 061, 19 pp.

Citation in format AMSBIB
\Bibitem{IsmKoe12} \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 \mathnet{http://mi.mathnet.ru/sigma738} \crossref{https://doi.org/10.3842/SIGMA.2012.061} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2970767} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000308737000001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84867021940} 

• http://mi.mathnet.ru/eng/sigma738
• http://mi.mathnet.ru/eng/sigma/v8/p61

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Groenevelt W., Ismail M.E.H., Koelink E., “Spectral Decomposition and Matrix-Valued Orthogonal Polynomials”, Adv. Math., 244 (2013), 91–105
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
4. F. A. Gruenbaum, L. Vinet, A. Zhedanov, “Tridiagonalization and the Heun equation”, J. Math. Phys., 58:3 (2017), 031703
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
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
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
8. Grunbaum F.A., Vinet L., Zhedanov A., “Algebraic Heun Operator and Band-Time Limiting”, Commun. Math. Phys., 364:3 (2018), 1041–1068
9. Alhaidari A.D., “Series Solutions of Heun-Type Equation in Terms of Orthogonal Polynomials”, J. Math. Phys., 59:11 (2018), 113507
10. Vinet L., Zhedanov A., “The Heun Operator of Hahn-Type”, Proc. Amer. Math. Soc., 147:7 (2019), 2987–2998
•  Number of views: This page: 112 Full text: 19 References: 21