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Resonances for Slowly Varying Perturbations of a Periodic Schrödinger Operator

Published online by Cambridge University Press:  20 November 2018

Mouez Dimassi*
Affiliation:
Université de Paris-Nord, Département de Mathématiques, UMR 7539, Institut Galilée, 93430 Villetaneuse, France
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Abstract

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We study the resonances of the operator $P(h)\,=\,-{{\Delta }_{x}}\,+\,V(x)\,+\,\varphi (hx)$. Here $V$ is a periodic potential, $\varphi $ a decreasing perturbation and $h$ a small positive constant. We prove the existence of shape resonances near the edges of the spectral bands of ${{P}_{0\,}}=\,-{{\Delta }_{x}}\,+\,V(x)$, and we give its asymptotic expansions in powers of ${{h}^{\frac{1}{2}}}$.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Aguilar, J. and Combes, J. M., A class of analytic perturbations for one-body Schrödinger Hamiltonians. Comm. Math. Phys. 22 (1971), 269279.Google Scholar
[2] Adams, E. N., Motion of electron in a perturbed periodic potential. Phys. Rev. 84 (1952), 4150.Google Scholar
[3] Balslev, E. and Combes, J. M., Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions. Comm. Math. Phys. 22 (1971), 280294.Google Scholar
[4] Bassani, F., Theory of imperfect crystalline solids. Triest lectures, Vienna, IAEA, 1970, 265.Google Scholar
[5] Bentosela, F., Scattering from impurities in a crystal. Comm. Math. Phys. 46 (1976), 153166.Google Scholar
[6] Birman, M., Discrete spectrum in the gaps of a continuous one for perturbations with large coupling limit. Adv. Soviet Math. 7 (1991), 5773.Google Scholar
[7] Briet, Ph., Combes, J. M. and Duclos, P., On the location of resonances for Schrödinger operators in the semiclassical limit. II. Barrier top resonances. Comm. Partial Differential Equations (2) 12 (1987), 201222.Google Scholar
[8] Brummelhuis, R. and Laguel, M., Résonances en limites semi-classique. Preprint, 2000.Google Scholar
[9] Buslaev, V. S., Semiclassical approximation for equations with periodic coefficients. Russian Math. Surveys, 42 (1987), 97125.Google Scholar
[10] Cycon, H. L., Resonances defined by modified dilations. Helv. Phys. Acta 58 (1985), 969981.Google Scholar
[11] Dimassi, M., Développements asymptotiques des perturbations lentes de l'opérateur de Schrödinger périodique. Comm. Partial Differential Equations 18 (1993), 771803.Google Scholar
[12] Dimassi, M. and Sjöstrand, J., Spectral asymptotics in the semi-classical limit. London Math. Soc. Lecture Note Ser. 268, Cambridge University Press, Cambridge, 1999.Google Scholar
[13] Firsova, N. E., Resonances of Hill operator perturbed by an exponentially decreasing additive potential. Math. Notes 36 (1984), 854861.Google Scholar
[14] Gérard, C., Resonance theory for periodic Schrödinger operators. Bull. Soc. Math. France 118 (1990), 2754.Google Scholar
[15] Gérard, C., Martinez, A. and Sjöstrand, J., A mathematical approach to the effective Hamiltonian in perturbed periodic problems. Comm. Math. Phys. 142 (1991), 217244.Google Scholar
[16] Gérard, C. and Nier, F., Scattering theory for the perturbations of periodic Schrödinger operators. J. Math. Kyoto Univ. 38 (1998), 595634.Google Scholar
[17] Guillot, J. C., Ralston, J. and Trubowitz, E., Semi-classical methods in solid state physics. Comm.Math. Phys. 116 (1988), 401415.Google Scholar
[18] Helffer, B. and Sjöstrand, J., On diamagnetism and de Haas-van Alphen effect. Ann. Inst. H. Poincar é Phys. Théor. 52 (1990), 303375.Google Scholar
[19] Helffer, B. and Sjöstrand, J., Analyse semi-classique pour l'équation de Harper. Mém. Soc. Math. France 34, 1988.Google Scholar
[20] Helffer, B. and Sjöstrand, J., Résonances en limite semi-classique. Mém. Soc. Math. France (N.S.) No. 24–25, 1986.Google Scholar
[21] Hörmander, L., The analysis of linear partial differential operator I. Springer-Verlag, Berlin-Heidelberg-New York, 1983.Google Scholar
[22] Hunziker, W., Distortion analyticity and molecular resonance curves. Ann. Inst. H. Poincar é Phys. Théor. 45 (1986), 339358.Google Scholar
[23] Kato, T., Perturbation theory for linear operators. Springer-Verlag, New York, 1966.Google Scholar
[24] Klein, M., Martinez, A. and Wang, X. P., On the Born-Oppenheimer approximation of wave operators in molecular scattering theory. Comm. Math. Phys. 152 (1993), 7395.Google Scholar
[25] Klopp, F., Resonances for perturbations of a semiclassical periodic Schrödinger operator. Ark. Mat. 52 (1994), 323371.Google Scholar
[26] Kuchment, P., Floquet theory for partial differential equations. Operator Theory: Advances and Applications 60, Birkhäuser Verlag, Basel, 1993.Google Scholar
[27] Kuchment, P. and Vainberg, B., On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials. Comm. Partial Differential Equations 25 (2000), 18091826.Google Scholar
[28] Luttinger, J. M., The effect of a magnetic field on electrons in a periodic potential. Phys. Rev. 84 (1951), 814817.Google Scholar
[29] Martinez, A., Résonances dans l'approximation de Born-Oppenheimer I. J. Differential Equations 91 (1991), 204234.Google Scholar
[30] Nakamura, S., Shape resonances for distortions analytic Schrödinger operators. Comm. Partial Differential Equations 14 (1989), 13851419.Google Scholar
[31] Reed, M. and Simon, B., Methods of modern mathematical physics. I. Functional analysis. Academic Press, New York-London, 1972.Google Scholar
[32] Reed, M. and Simon, B., Methods of modern mathematical physics, IV. Analysis operators. Academic Press, New York, (1978).Google Scholar
[33] Robert, D., Autour de l'approximation semiclassique. Birkhäuser, Basel, 1983.Google Scholar
[34] Sjöstrand, J., Microlocal analysis for periodic magnetic equation and related questions. Lecture Notes in Math. 1495, Springer-Verlag, Berlin-Heidelberg, 1991, 237332.Google Scholar
[35] Sjöstrand, J., Resonances and semiclassical analysis. Lecture Notes in Phys. 325, Springer, Berlin-New York, 1989, 2133.Google Scholar
[36] Sjöstrand, J. and Zworski, M., Asymptotic distribution of resonances for convex obstacles. Acta Math. 183 (1999), 191253.Google Scholar
[37] Slater, J. C., Electron in perturbed periodic lattices. Phys. Rev. 76 (1949), 15921601.Google Scholar
[38] Wang, X. P., Barrier resonances in strong magnetic fields. Comm. Partial Differential Equations 17 (1992), 15391566.Google Scholar