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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 1, Issue 4 (Dec 2003)


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

Michael Melgaard
  • Chalmers University of Technology
  • Email:
Published Online: 2003-12-01 | DOI: https://doi.org/10.2478/BF02475180


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.

Keywords: Near-threshold resolvent expansions; scattering matrix; auxiliary one-dimensional Schrödinger operator

Keywords: 47N20; 35J10 35P25 47F05 81U05

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

  • [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)90242-7 [Crossref]

  • [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.

  • [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.

  • [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/S0012-7094-78-04540-4 [Crossref]

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

  • [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.

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

  • [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)90281-2

  • [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)90032-7 [Crossref]

  • [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.

  • [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/10.1063/1.526314 [Crossref]

  • [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.

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

  • [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.

  • [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/S0012-7094-79-04631-3 [Crossref]

  • [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.

  • [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.

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

  • [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.

  • [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/BF01942369 [Crossref]

  • [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/10.1088/0305-4470/28/12/021 [Crossref]

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

  • [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/S0012-7094-93-07005-6 [Crossref]

  • [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-X [Crossref]

  • [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-D

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

  • [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.2000.7325 [Crossref]

  • [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/BF01942331 [Crossref]

  • [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/10.1063/1.523428 [Crossref]

  • [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/10.1063/1.523466 [Crossref]

  • [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.

  • [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/10.2307/1971301 [Crossref]

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

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

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

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

  • [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.

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

About the article

Published Online: 2003-12-01

Published in Print: 2003-12-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/BF02475180. Export Citation

© 2003 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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