Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 1, 2003

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

  • Michael Melgaard EMAIL logo
From the journal Open Mathematics

Abstract

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H m+Hom+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H m, H om) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.

[1] M. Abramovitz and I. Stegun: Handbook of Mathematical Functions, Dover Publications. New York, 1972. Search in Google Scholar

[2] S. Albeverio, D. Bollé, F. Gesztesy, R. Hoegh-Krohn: “Low-energy parameters in nonrelativistic scattering theory”, Ann. Physics, Vol. 148, (1983), pp.308–326. http://dx.doi.org/10.1016/0003-4916(83)90242-710.1016/0003-4916(83)90242-7Search in Google Scholar

[3] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn: “The low energy expansion in nonrelativistic scattering theory”, Ann. Inst. H. Poincaré Sect. A (N.S.), Vol. 37, (1982), pp. 1–28. Search in Google Scholar

[4] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, L. Streit: “Charged particles with short range interactions”, Ann. Inst. H. Poincaré Sect. A (N.S.), Vol. 38, (1983), pp. 263–293. Search in Google Scholar

[5] J.E. Avron, I. Herbst, B. Simon: “Schrödinger operators with magnetic fields. I. General interactions”, Duke Math. J., Vol. 45, (1978), pp. 847–883. http://dx.doi.org/10.1215/S0012-7094-78-04540-410.1215/S0012-7094-78-04540-4Search in Google Scholar

[6] Ju. M. Berezanskii: Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monographs Vol. 17, American Mathematical Society, Providence, Rhode Island, 1968. Search in Google Scholar

[7] D. Bollé: “Schrödinger operators at threshold”, In: S. Albeverio, J.E. Fenstad, H. Holden, T. Lindström (Eds.): Ideas and Methods in Quantum and Statistical Physics (Oslo, 1988), Cambridge University Press, Cambridge, 1992, pp. 173–196. Search in Google Scholar

[8] D. Bollé, F. Gesztesy, C. Danneels: “Threshold scattering in two dimensions”, Ann. Inst. H. Poincaré Phys. Théor., Vol. 48, (1988), pp. 175–204. Search in Google Scholar

[9] D. Bollé, F. Gesztesy, M. Klaus: “Scattering theory for one-dimensional systems with ∫dx V (x)=0”, J. Math. Anal. Appl., Vol.122, (1987) pp. 496–518. http://dx.doi.org/10.1016/0022-247X(87)90281-210.1016/0022-247X(87)90281-2Search in Google Scholar

[10] D. Bollé, F. Gesztesy, C. Nessmann, L. Streit: “Scattering theory for general, nonlocal interactions: threshold behavior and sum rules”, Rep. Math. Phys., Vol. 23, (1986), pp. 373–408. http://dx.doi.org/10.1016/0034-4877(86)90032-710.1016/0034-4877(86)90032-7Search in Google Scholar

[11] D. Bollé, F. Gesztesy, S.F.J. Wilk: “A complete treatment of low-energy scattering in one dimension”, J. Operator Theory, Vol. 13, (1985), pp. 3–31. Search in Google Scholar

[12] M. Cheney: “Two-dimensional scattering: the number of bound states from scattering data”, J. Math. Phys., Vol. 25, (1984), pp. 1449–1455. http://dx.doi.org/10.1063/1.52631410.1063/1.526314Search in Google Scholar

[13] H.L. Cycon, R.G. Froese, W. Kirch, B. Simon: Schrödinger Operators—With Applications to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, Heidelberg, New York, 1987. 10.1007/978-3-540-77522-5Search in Google Scholar

[14] P.G. Dodds and D.H. Fremlin: “Compact operators in Banach lattices”, Israel J. Math., Vol. 34, (1979), pp. 287–320. Search in Google Scholar

[15] P.D. Hislop and I.M. Sigal: Introduction to Spectral Theory. With applications to Schrödinger operators, Applied Mathematical Sciences 113, Springer-Verlag New York, Inc., 1996. 10.1007/978-1-4612-0741-2Search in Google Scholar

[16] A. Jensen and T. Kato: “Spectral properties of Schrödinger operators and time-decay of the wave functions”, Duke Math. J., Vol. 46, (1979), pp. 583–611. http://dx.doi.org/10.1215/S0012-7094-79-04631-310.1215/S0012-7094-79-04631-3Search in Google Scholar

[17] A. Jensen and M. Melgaard: “Perturbation of eigenvalues embedded at a threshold”, Proc. Roy. Soc. Edinburgh Sect., Vol. 131 A, (2002), pp. 163–179. Search in Google Scholar

[18] A. Jensen, E. Mourre, P. Perry: “Multiple commutator estimates and resolvent smoothness in quantum scattering theory”, Ann. Inst. Henri Poincaré, Vol. 41, (1984), pp. 207–225. Search in Google Scholar

[19] A. Jensen: “Scattering theory for Stark Hamiltonians”, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 104, (1994), pp. 599–651. Search in Google Scholar

[20] T. Kato: Perturbation Theory for Linear Operators, Volume 132 of Die Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, Heidelberg, New York, second edition, 1976. Search in Google Scholar

[21] M. Klaus and B. Simon: “Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case”, Commun. Math. Phys., Vol. 78, (1980), pp. 251–281. http://dx.doi.org/10.1007/BF0194236910.1007/BF01942369Search in Google Scholar

[22] V.V. Kostrykin, A.A. Kvitsinsky, S.P. Merkuriev: “Potential scattering in a constant magnetic field: spectral asymptotics and Levinson formula”, J. Phys. A: Math. Gen., Vol. 28, (1995), pp. 3493–3509. http://dx.doi.org/10.1088/0305-4470/28/12/02110.1088/0305-4470/28/12/021Search in Google Scholar

[23] S.T. Kuroda: An Introduction to Scattering Theory, Aarhus University, Matematisk Institut, Lecture Notes Series, 1978. Search in Google Scholar

[24] I. Laba: “Long-range one-particle scattering in a homogeneous magnetic field”, Duke Math. J., Vol. 70, (1993), pp. 283–303. http://dx.doi.org/10.1215/S0012-7094-93-07005-610.1215/S0012-7094-93-07005-6Search in Google Scholar

[25] M.R.C. McDowell and M. Zarcone: “Scattering in strong magnetic fields”, Adv.At. Mol. Phys., Vol. 21, (1986), pp. 255–304. http://dx.doi.org/10.1016/S0065-2199(08)60144-X10.1016/S0065-2199(08)60144-XSearch in Google Scholar

[26] M. Melgaard: “Spectral properties in the low-energy limit of one-dimensional Schrödinger operators. The case 〈1, V1〉≠0”, Math. Nachr., Vol. 238, (2002), pp. 113–143. http://dx.doi.org/10.1002/1522-2616(200205)238:1<113::AID-MANA113>3.0.CO;2-D10.1002/1522-2616(200205)238:1<113::AID-MANA113>3.0.CO;2-DSearch in Google Scholar

[27] M. Melgaard: “Spectral properties in the low-energy limit of one-dimensional Schrödinger operators. The case 〈1, V1〉=0”, in preparation. Search in Google Scholar

[28] M. Melgaard: “Spectral properties at a threshold for two-channel Hamiltonians. I. Abstract theory”, J. Math. Anal. Appl., Vol. 256, (2001), pp. 281–303. http://dx.doi.org/10.1006/jmaa.2000.732510.1006/jmaa.2000.7325Search in Google Scholar

[29] E. Mourre: “Absence of singular continuous spectrum for certain self-adjoint operators”, Commun. Math. Phys., Vol. 78, (1981), pp. 391–408. http://dx.doi.org/10.1007/BF0194233110.1007/BF01942331Search in Google Scholar

[30] R.G. Newton: “Noncentral potentials: the generalized Levinson theorem and the structure of the spectrum”, J. Math. Phys., Vol. 18, (1977), pp. 1348–1357. http://dx.doi.org/10.1063/1.52342810.1063/1.523428Search in Google Scholar

[31] R.G. Newton: “Nonlocal interactions; the generalized Levinson theorem and the structure of the spectrum”, J. Math. Phys., Vol. 18, (1977), pp. 1582–1588. http://dx.doi.org/10.1063/1.52346610.1063/1.523466Search in Google Scholar

[32] R.G. Newton: “Bounds on the number of bound states for the Schrödinger equation in one and two dimensions”, J. Operator Theory, 10, (1983), pp. 119–125. Search in Google Scholar

[33] P. Perry, I.M. Sigal, B. Simon: “Spectral analysis of N-body Schrödinger operators”, Ann. Math., Vol. 114, (1981), pp. 516–567. http://dx.doi.org/10.2307/197130110.2307/1971301Search in Google Scholar

[34] L. Pitt: “A compactness condition for linear operators in function spaces”, J. Operator Theory, Vol. 1, (1979) pp. 49–54. Search in Google Scholar

[35] M. Reed and B. Simon: Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press Inc., 1975. Search in Google Scholar

[36] M. Reed and B. Simon: Methods of modern mathematical physics. IV: Analysis of operators, Academic Press Inc., London, 1978. Search in Google Scholar

[37] M. Schechter: Spectra of Partial Differential Operators, First edition, North-Holland, Amsterdam, New York, Oxford, 1971. Search in Google Scholar

[38] S.N. Solnyshkin: “Asymptotics of the energy of bound states of the Schrödinger operator in the presence of electric and homogeneous magnetic fields”, Sel. Math. Sov., Vol. 5, (1986), pp. 297–306. Search in Google Scholar

[39] H. Tamura, “Magnetic scattering at low energy in two dimensions”, Nagoya Math. J., Vol. 155, (1999), pp. 95–151. Search in Google Scholar

Published Online: 2003-12-1
Published in Print: 2003-12-1

© 2003 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.2478/BF02475180/html
Scroll to top button