Eigensystem of an L 2-perturbed harmonic oscillator is an unconditional basis

James Adduci ; Boris Mityagin

Open Mathematics, Tome 10 (2012), p. 569-589 / Harvested from The Polish Digital Mathematics Library

Résumé

For any complex valued L p-function b(x), 2 ≤ p < ∞, or L ∞-function with the norm ‖b↾L ∞‖ < 1, the spectrum of a perturbed harmonic oscillator operator L = −d 2/dx 2 + x 2 + b(x) in L 2(ℝ1) is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in L 2(ℝ).

Détail

Publié le : 2012-01-01

Zbl 1259.47059

EUDML-ID : urn:eudml:doc:269402

Bibliographie

[1] Agranovich M.S., Elliptic Operators on Closed Manifolds, Encyclopaedia Math. Sci., 63, Springer, Berlin-Heidelberg, 1994

[2] Agranovich M.S., Katsenelenbaum B.Z., Sivov A.N., Voitovich N.N., Generalized Method of Eigenoscillations in Diffraction Theory, Wiley-VCH, Berlin, 1999 | Zbl 0929.65097

[3] Akhmerova É.F., The asymptotics of the spectrum of nonsmooth perturbations of a harmonic oscillator, Sib. Math. J., 2008, 49(6), 968–984 http://dx.doi.org/10.1007/s11202-008-0093-x | Zbl 1217.47028

[4] Albeverio S., Motovilov A.K., Shkalikov A.A., Bounds on variation of spectral subspaces under J-self-adjoint perturbations, Integral Equations Operator Theory, 2009, 64(4), 455–486 http://dx.doi.org/10.1007/s00020-009-1702-1 | Zbl 1197.47024

[5] Askey R., Wainger S., Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math., 1965, 87(3), 695–708 http://dx.doi.org/10.2307/2373069 | Zbl 0125.31301

[6] Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys., 2007, 70(6), 947–1018 http://dx.doi.org/10.1088/0034-4885/70/6/R03

[7] Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett., 1998, 80(24), 5243–5246 http://dx.doi.org/10.1103/PhysRevLett.80.5243 | Zbl 0947.81018

[8] Caliceti E., Cannata F., Graffi S., PT symmetric Schrödinger operators: reality of the perturbed eigenvalues, SIGMA Symmetry Integrability Geom. Methods Appl., 2010, 6, #8 | Zbl 1202.47014

[9] Chelkak D.S., Approximation in the space of spectral data of a perturbed harmonic oscillator, J. Math. Sci. (N. Y.), 2003, 117(3), 4260–4269 http://dx.doi.org/10.1023/A:1024824821966

[10] Chelkak D., Kargaev P., Korotyaev E., An inverse problem for an harmonic oscillator perturbed by potential: uniqueness, Lett. Math. Phys., 2003, 64(1), 7–21 http://dx.doi.org/10.1023/A:1024985302559 | Zbl 1030.34085

[11] Chelkak D., Kargaev P., Korotyaev E., Inverse problem for harmonic oscillator perturbed by potential, characterization, Comm. Math. Phys., 2004, 249(1), 133–196 http://dx.doi.org/10.1007/s00220-004-1105-8 | Zbl 1085.81055

[12] Chelkak D., Kargaev P., Korotyaev E., Inverse problem for harmonic oscillator perturbed by potential, In: Inverse Problems and Spectral Theory, Kyoto, October 28–November 1, 2002, Contemp. Math., 348, American Mathematical Society, Providence, 2004, 93–102 | Zbl 1073.34013

[13] Chelkak D., Korotyaev E., The inverse problem for perturbed harmonic oscillator on the half-line with a Dirichlet boundary condition, Ann. Henri Poincaré, 2007, 8(6), 1115–1150 http://dx.doi.org/10.1007/s00023-007-0330-z | Zbl 1130.81036

[14] Cramér H., On some classes of series used in mathematical statistics, In: Proceedings of the Sixth Scandinavian Congress of Mathematicians, Copenhagen, 1926, 399–425 | Zbl 52.0518.01

[15] Dirac P., The Principles of Quantum Mechanics, 4th ed., Internat. Ser. Monogr. Phys., Oxford University Press, London, 1958 | Zbl 0080.22005

[16] Djakov P., Mityagin B., Bari-Markus property for Riesz projections of Hill operators with singular potentials, In: Functional Analysis and Complex Analysis, Sabanc? University, İstanbul, September 17–21, 2007, Contemp. Math., 481, American Mathematical Society, Providence, 2009, 59–80 | Zbl 1185.34130

[17] Djakov P., Mityagin B., Bari-Markus property for Riesz projections of 1D periodic Dirac operators, Math. Nach., 2010, 283(3), 443–462 http://dx.doi.org/10.1002/mana.200910003 | Zbl 1198.34188

[18] Erdélyi A., Asymptotic solutions of differential equations with transition points or singularities, J. Mathematical Phys., 1960, 1(1), 16–26 http://dx.doi.org/10.1063/1.1703631 | Zbl 0125.04802

[19] Gohberg I.C., Krełn M.G., Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., 18, American Mathematical Society, Providence, 1969

[20] Hardy G.H., Littlewood J.E., Pólya G., Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952

[21] Hille E., A class of reciprocal functions, Ann. of Math., 1926, 27(4), 427–464 http://dx.doi.org/10.2307/1967695 | Zbl 52.0400.02

[22] Hruščëv S.V., Nikol’skił N.K., Pavlov B.S., Unconditional bases of exponentials and of reproducing kernels, In: Complex Analysis and Spectral Theory, Leningrad, 1979/80, Lecture Notes in Math., 864, Springer, Berlin-Heidelberg-New York, 1981, 214–335

[23] Hunt R., Muckenhoupt B., Wheeden R., Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc., 1973, 176, 227–251 http://dx.doi.org/10.1090/S0002-9947-1973-0312139-8 | Zbl 0262.44004

[24] Kato T., Similarity for sequences of projections, Bull. Amer. Math. Soc., 1967, 73(6), 904–905 http://dx.doi.org/10.1090/S0002-9904-1967-11836-6 | Zbl 0156.38103

[25] Kato T., Perturbation Theory for Linear Operators, 2nd ed. (reprint), Classics Math., Springer, Berlin, 1995

[26] Katsnelson V.E., Conditions for a system of root vectors of certain classes of operators to be a basis, Funkcional. Anal. i Prilozhen., 1967, 1(2), 39–51 (in Russian)

[27] Levitan B.M., Sargsjan I.S., Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Transl. Math. Monogr., 39, American Mathematical Society, Providence, 1975 | Zbl 0302.47036

[28] Markus A.S., A basis of root vectors of a dissipative operator, Dokl. Akad. Nauk SSSR, 1960, 132, 524–527 (in Russian)

[29] Markus A.S., Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Monogr., 71, American Mathematical Society, Providence, 1988

[30] Mityagin B.S., Convergence of expansions in eigenfunctions of the Dirac operator, Dokl. Akad. Nauk, 2003, 393(4), 456–459 (in Russian)

[31] Mityagin B., Spectral expansions of one-dimensional periodic Dirac operators, Dyn. Partial Differ. Equ., 2004, 1(2), 125–191

[32] Mostafazadeh A., Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys., 2010, 7(7), 1191–1306 http://dx.doi.org/10.1142/S0219887810004816 | Zbl 1208.81095

[33] Nevai P., Exact bounds for orthogonal polynomials associated with exponential weights, J. Approx. Theory, 1985, 44(1), 82–85 http://dx.doi.org/10.1016/0021-9045(85)90070-X | Zbl 0605.42019

[34] Savchuk A.M., Shkalikov A.A., Sturm-Liouville operators with distribution potentials, Tr. Mosk. Mat. Obs., 2003, 64, 159–212 (in Russian) | Zbl 1066.34085

[35] Skovgaard H., Asymptotic forms of Hermite polynomials, Technical report, 18, Department of Mathematics, California Institute of Technology, Pasadena, 1959

[36] Szegö G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., 23, American Mathematical Society, New York, 1939

[37] Thangavelu S., Lectures on Hermite and Laguerre Expansions, Math. Notes, 42, Princeton University Press, Princeton, 1993

[38] Tkachenko V., Non-selfadjoint Sturm-Liouville operators with multiple spectra, In: Interpolation Theory, Systems Theory and Related Topics, Tel Aviv/Rehovot, 1999, Oper. Theory Adv. Appl., 134, Birkhäuser, Basel, 2002, 403–414 http://dx.doi.org/10.1007/978-3-0348-8215-6_17

[39] Znojil M., Non-Hermitian supersymmetry and singular, PT -symmetrized oscillators, J. Phys. A, 2002, 35(9), 2341–2352 http://dx.doi.org/10.1088/0305-4470/35/9/320

[40] Zygmund A., Trigonometric Series, 2nd ed., Cambridge University Press, London-New York, 1968 | Zbl 0157.38204