Spectral analysis of certain Schrödinger operators

Mourad E.H. Ismail ; Erik Koelink

Symmetry, integrability and geometry: methods and applications, Tome 8 (2012)

Résumé

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].

Détail

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.

Bibliographie

[1] Akhiezer N.I., The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965 | MR

[2] Alhaidari A.D., “An extended class of $L^2$-series solutions of the wave equation”, Ann. Physics, 317 (2005), 152–174 ; arXiv: quant-ph/0409002 | DOI | MR | Zbl

[3] Alhaidari A.D., “Exact $L^2$ series solution of the Dirac–Coulomb problem for all energies”, Ann. Physics, 312 (2004), 144–160 ; arXiv: hep-th/0405023 | DOI | MR | Zbl

[4] Alhaidari A.D., “Group-theoretical foundation of the {$J$}-matrix theory of scattering”, J. Phys. A: Math. Gen., 33 (2000), 6721–6738 | DOI | MR | Zbl

[5] Alhaidari A.D., “$L^2$ series solution of the relativistic Dirac–Morse problem for all energies”, Phys. Lett. A, 326 (2004), 58–69 ; arXiv: math-ph/0405008 | DOI | MR | Zbl

[6] Alhaidari A.D., Bahlouli H., “Extending the class of solvable potentials. I. The infinite potential well with a sinusoidal bottom”, J. Math. Phys., 49 (2008), 082102, 13 pp. | DOI | MR | Zbl

[7] Alhaidari A.D., Bahlouli H., Abdelmonem M.S., Al-Ameen F., Al-Abdulaal T., “Regularization in the {$J$}-matrix method of scattering revisited”, Phys. Lett. A, 364 (2007), 372–377 | DOI | MR | Zbl

[8] Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[9] Askey R., “Continuous $q$-Hermite polynomials when $q>1$”, $q$-Series and Partitions (Minneapolis, MN, 1988), IMA Vol. Math. Appl., 18, Springer, New York, 1989, 151–158 | DOI | MR

[10] Bastard G., Wave mechanics applied to semiconductor heterostructures, Wiley-Interscience, 1991

[11] Berezans'kiĭ J.M., Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, 17, American Mathematical Society, Providence, RI, 1968 | MR | Zbl

[12] Brown B.M., Evans W.D., Ismail M.E.H., “The Askey–Wilson polynomials and $q$-Sturm–Liouville problems”, Math. Proc. Cambridge Philos. Soc., 119 (1996), 1–16 ; arXiv: math.CA/9408209 | DOI | MR | Zbl

[13] Christiansen J.S., Koelink E., “Self-adjoint difference operators and symmetric Al-Salam–Chihara polynomials”, Constr. Approx., 28 (2008), 199–218 ; arXiv: math.CA/0610534 | DOI | MR | Zbl

[14] Deift P.A., Orthogonal polynomials and random matrices: a Riemann–Hilbert approach, Courant Lecture Notes in Mathematics, 3, New York University Courant Institute of Mathematical Sciences, New York, 1999 | MR

[15] Edmunds D.E., Evans W.D., Spectral theory and differential operators, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1987 | MR

[16] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, v. 1, McGraw-Hill, 1953

[17] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[18] Goh S.S., Micchelli C.A., “Uncertainty principles in Hilbert spaces”, J. Fourier Anal. Appl., 8 (2002), 335–373 | DOI | MR | Zbl

[19] Heller E.J., “Theory of $J$-matrix Green's functions with applications to atomic polarizability and phase-shift error bounds”, Phys. Rev. A, 12 (1975), 1222–1231 | DOI

[20] Heller E.J., Reinhardt W.P., Yamani H.A., “On an “equivalent quadrature” calculation of matrix elements of $(z-P^2/2m)$ using an $L^2$ expansion technique”, J. Comp. Phys., 13 (1973), 536–549 | DOI

[21] Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, 98, Cambridge University Press, Cambridge, 2009 | MR | Zbl

[22] Ismail M.E.H., “Ladder operators for $q^{-1}$-Hermite polynomials”, C.R. Math. Rep. Acad. Sci. Canada, 15 (1993), 261–266 | MR | Zbl

[23] Ismail M.E.H., Koelink E., “Spectral properties of operators using tridiagonalization”, Anal. Appl., 10 (2012), 327–343 ; arXiv: 1108.5716 | DOI | Zbl

[24] Ismail M.E.H., Koelink E., “The $J$-matrix method”, Adv. in Appl. Math., 46 (2011), 379–395 ; arXiv: 0810.4558 | DOI | MR | Zbl

[25] Ismail M.E.H., Letessier J., Masson D., Valent G., “Birth and death processes and orthogonal polynomials”, Orthogonal Polynomials: Theory and Practice (Columbus, OH, 1989), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 294, ed. P. Nevai, Kluwer Acad. Publ., Dordrecht, 1990, 225–229 | MR

[26] Ismail M.E.H., Li X., “Bound on the extreme zeros of orthogonal polynomials”, Proc. Amer. Math. Soc., 115 (1992), 131–140 | DOI | MR | Zbl

[27] Ismail M.E.H., Masson D.R., “$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals”, Trans. Amer. Math. Soc., 346 (1994), 63–116 | DOI | MR | Zbl

[28] Karlin S., Tavaré S., “Linear birth and death processes with killing”, J. Appl. Probab., 19 (1982), 477–487 | DOI | MR | Zbl

[29] Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998; http://aw.twi.tudelft.nl/~koekoek/askey/

[30] Koelink E., “Spectral theory and special functions”, Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY, 2004, 45–84 ; arXiv: math.CA/0107036 | MR | Zbl

[31] Koelink H.T., Van der Jeugt J., “Convolutions for orthogonal polynomials from Lie and quantum algebra representations”, SIAM J. Math. Anal., 29 (1998), 794–822 ; arXiv: q-alg/9607010 | DOI | MR | Zbl

[32] Landau L.D., Lifshitz L.M., Course of theoretical physics, v. 3, Quantum mechanics: nonrelativistic theory, 3rd ed., Butterworth-Heinemann, 1991

[33] Masson D.R., Repka J., “Spectral theory of Jacobi matrices in $l^2(\mathbf Z)$ and the $\mathrm{su}(1,1)$ Lie algebra”, SIAM J. Math. Anal., 22 (1991), 1131–1146 | DOI | MR | Zbl

[34] Miller Jr. W., Lie theory and special functions, Mathematics in Science and Engineering, 43, Academic Press, New York, 1968 | DOI | MR | Zbl

[35] Odake S., Sasaki R., “Discrete quantum mechanics”, J. Phys. A: Math. Theor., 44 (2011), 353001, 47 pp. ; arXiv: 1104.0473 | DOI | MR | Zbl

[36] Ojha P.C., “$\mathrm{SO}(2,1)$ Lie algebra, the Jacobi matrix and the scattering states of the Morse oscillator”, J. Phys. A: Math. Gen., 21 (1988), 875–883 | DOI | MR | Zbl

[37] Rainville E.D., Special functions, The Macmillan Co., New York, 1960 | MR | Zbl

[38] Simon B., “The classical moment problem as a self-adjoint finite difference operator”, Adv. Math., 137 (1998), 82–203 ; arXiv: math-ph/9906008 | DOI | MR | Zbl

[39] Szeg{ő} G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, 23, 4th ed., American Mathematical Society, Providence, RI, 1975 | MR

[40] Yamani H.A., Reinhardt W.P., “$L^2$ discretizations of the continuum: radial kinetic energy”, Phys. Rev. A, 11 (1975), 1144–1156 | DOI