Resonances and Spectral Shift Function near the Landau levels

Jean-François Bony[1]; Vincent Bruneau[1]; Georgi Raikov[2]

  • [1] Université Bordeaux I FR CNRS 2254, MAB UMR CNRS 5466 Institut de Mathématiques de Bordeaux 351 cours de la Libération 33405 Talence (France)
  • [2] Pontificia Universidad Católica de Chile Facultad de Matemáticas Departamento de Matemáticas Vicuña Mackenna 4860 Santiago de Chile (Chile)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 2, page 629-671
  • ISSN: 0373-0956

Abstract

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We consider the 3D Schrödinger operator H = H 0 + V where H 0 = ( - i ∇ - A ) 2 - b , A is a magnetic potential generating a constant magneticfield of strength b > 0 , and V is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of H admits a meromorphic extension from the upper half plane to an appropriate Riemann surface ℳ , and define the resonances of H as the poles of this meromorphic extension. We study their distribution near any fixed Landau level 2 b q , q ∈ ℕ . First, we obtain a sharp upper bound of the number of resonances in a vicinity of 2 b q . Moreover, under appropriate hypotheses, we establish corresponding lower bounds which imply the existence of an infinite number of resonances, or the absence of resonances in certain sectors adjoining 2 b q . Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair ( H , H 0 ) as a sum of a harmonic measure related to the resonances, and the imaginary part of a holomorphic function. This representation justifies the Breit-Wigner approximation, implies a trace formula, and provides information on the singularities of the SSF at the Landau levels.

How to cite

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Bony, Jean-François, Bruneau, Vincent, and Raikov, Georgi. "Resonances and Spectral Shift Function near the Landau levels." Annales de l’institut Fourier 57.2 (2007): 629-671. <http://eudml.org/doc/10234>.

@article{Bony2007,
abstract = {We consider the 3D Schrödinger operator $H = H_0 + V$ where $H_0 = (-i\nabla - A)^2 -b$, $A$ is a magnetic potential generating a constant magneticfield of strength $b&gt;0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a meromorphic extension from the upper half plane to an appropriate Riemann surface $\{\mathcal\{M\}\}$, and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed Landau level $2bq$, $q \in \{\mathbb\{N\}\}$. First, we obtain a sharp upper bound of the number of resonances in a vicinity of $2bq$. Moreover, under appropriate hypotheses, we establish corresponding lower bounds which imply the existence of an infinite number of resonances, or the absence of resonances in certain sectors adjoining $2bq$. Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair $(H,H_0)$ as a sum of a harmonic measure related to the resonances, and the imaginary part of a holomorphic function. This representation justifies the Breit-Wigner approximation, implies a trace formula, and provides information on the singularities of the SSF at the Landau levels.},
affiliation = {Université Bordeaux I FR CNRS 2254, MAB UMR CNRS 5466 Institut de Mathématiques de Bordeaux 351 cours de la Libération 33405 Talence (France); Université Bordeaux I FR CNRS 2254, MAB UMR CNRS 5466 Institut de Mathématiques de Bordeaux 351 cours de la Libération 33405 Talence (France); Pontificia Universidad Católica de Chile Facultad de Matemáticas Departamento de Matemáticas Vicuña Mackenna 4860 Santiago de Chile (Chile)},
author = {Bony, Jean-François, Bruneau, Vincent, Raikov, Georgi},
journal = {Annales de l’institut Fourier},
keywords = {Magnetic Schrödinger operators; resonances; spectral shift function; Breit-Wigner approximation; magnetic Schrödinger operators},
language = {eng},
number = {2},
pages = {629-671},
publisher = {Association des Annales de l’institut Fourier},
title = {Resonances and Spectral Shift Function near the Landau levels},
url = {http://eudml.org/doc/10234},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Bony, Jean-François
AU - Bruneau, Vincent
AU - Raikov, Georgi
TI - Resonances and Spectral Shift Function near the Landau levels
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 2
SP - 629
EP - 671
AB - We consider the 3D Schrödinger operator $H = H_0 + V$ where $H_0 = (-i\nabla - A)^2 -b$, $A$ is a magnetic potential generating a constant magneticfield of strength $b&gt;0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a meromorphic extension from the upper half plane to an appropriate Riemann surface ${\mathcal{M}}$, and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed Landau level $2bq$, $q \in {\mathbb{N}}$. First, we obtain a sharp upper bound of the number of resonances in a vicinity of $2bq$. Moreover, under appropriate hypotheses, we establish corresponding lower bounds which imply the existence of an infinite number of resonances, or the absence of resonances in certain sectors adjoining $2bq$. Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair $(H,H_0)$ as a sum of a harmonic measure related to the resonances, and the imaginary part of a holomorphic function. This representation justifies the Breit-Wigner approximation, implies a trace formula, and provides information on the singularities of the SSF at the Landau levels.
LA - eng
KW - Magnetic Schrödinger operators; resonances; spectral shift function; Breit-Wigner approximation; magnetic Schrödinger operators
UR - http://eudml.org/doc/10234
ER -

References

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  1. J. Avron, I. Herbst, B. Simon, Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J. 45 (1978), 847-883 Zbl0399.35029MR518109
  2. A. Sá Barreto, M. Zworski, Existence of resonances in three dimensions, Commun. Math. Phys. 173 (1995), 401-415 Zbl0835.35099MR1355631
  3. J.-F. Bony, J. Sjöstrand, Trace formula for resonances in small domains, J. Funct. Anal. 184 (2001), 402-418 Zbl1068.47055MR1851003
  4. J. M. Bouclet, Traces formulae for relatively Hilbert-Schmidt perturbations, Asymptot. Anal. 32 (2002), 257-291 Zbl1062.47021MR1993651
  5. J. M. Bouclet, Spectral distributions for long range perturbations, J. Funct. Anal. 212 (2004), 431-471 Zbl1088.35041MR2064934
  6. V. Bruneau, V. Petkov, Meromorphic continuation of the spectral shift function, Duke Math. J. 116 (2003), 389-430 Zbl1033.35081MR1958093
  7. V. Bruneau, A. Pushnitski, G. D. Raikov, Spectral shift function in strong magnetic fields, Algebra i Analysis 16 (2004), 207-238 Zbl1082.35115MR2069004
  8. D. Delande, A. Bommier, J.-C. Gay, Positive-Energy spectrum of the hydrogen atom in a magnetic field, Phys. Rev. Lett. 66 (1991), 141-144 
  9. M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, Lecture Notes Series, London Math. Society 268 (1999), Cambridge University Press Zbl0926.35002MR1735654
  10. M. Dimassi, M. Zerzeri, A local trace formula for resonances of perturbed periodic Schrödinger operators, J. Funct. Anal. 198 (2003), 142-159 Zbl1090.35065MR1962356
  11. C. Fernández, G. D. Raikov, On the singularities of the magnetic spectral shift function at the Landau levels, Ann. Henri Poincaré 5 (2004), 381-403 Zbl1062.81043MR2057679
  12. N. Filonov, A. Pushnitski, Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains, Commun. Math. Phys. 264 (2006), 759-772 Zbl1106.81040MR2217290
  13. R. Froese, Asymptotic distribution of resonances in one dimension, J. Diff. Equa. 137 (1997), 251-272 Zbl0955.35057MR1456597
  14. R. Froese, R. Waxler, Ground state resonances of a hydrogen atom in an intense magnetic field, Rev. Math. Phys. 7 (1995), 311-361 Zbl0836.47048MR1326138
  15. W. Fulton, Algebraic Topology, A First Course, Graduate Texts in Mathematics (1995), Springer Zbl0852.55001MR1343250
  16. I. C. Gohberg, M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs 18 (1969), American Math. Society, Providence, R.I. Zbl0181.13504MR246142
  17. L. S. Koplienko, Trace formula for non trace-class perturbations, Sibirsk. Mat. Zh. 25 (1984), 62-71 Zbl0574.47021MR762239
  18. L. S. Koplienko, Regularized function of spectral shift for a one-dimensional Schrödinger operator with slowly decreasing potential, Sibirsk. Mat. Zh. 26 (1985), 72-77, 62-71 Zbl0581.47034MR792056
  19. M. G. Krein, On perturbation determinants and a trace formula for unitary and self-adjoint operators, Dokl. Akad. Nauk SSSR 144 (1962), 268-271 Zbl0191.15201MR139006
  20. L. Landau, Diamagnetismus der Metalle, Z. Physik 64 (1930), 629-637 
  21. V. Petkov, M. Zworski, Semi-classical estimates on the scattering determinant, Ann. H. Poincaré 2 (2001), 675-711 Zbl1041.81041MR1852923
  22. G. D. Raikov, Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips, Commun. PDE 15 (1990), 407-434 Zbl0739.35055MR1044429
  23. G. D. Raikov, Spectral shift function for Schrödinger operators in constant magnetic fields, Cubo 7 (2005), 171-199 Zbl1103.81018MR2186031
  24. G. D. Raikov, Spectral shift function for magnetic Schrödinger operators, Mathematical Physics of Quantum Mechanics, Proceedings of the Conference Math. 9, Giens (France), 2004, Lecture Notes in Physics 690 (2006), 451-465, Springer Zbl1167.81383MR2235707
  25. G. D. Raikov, S. Warzel, Quasi-classical versus non-classical spectral asymptotics for magnetic Schödinger operators with decreasing electric potentials, Rev. Math. Phys. 14 (2002), 1051-1072 Zbl1033.81038MR1939760
  26. J. Sjöstrand, Lectures on resonances Zbl0702.35188
  27. J. Sjöstrand, A trace formula for resonances and application to semi-classical Schrödinger operator, Séminaire EDP, Exposé II, École Polytechnique (1996-1997), 1-17 Zbl1061.35506MR1482808
  28. J. Sjöstrand, A trace formula and review of some estimates for resonances, Microlocal analysis and spectral theory (Lucca, 1996), p.377–437, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 490 (1997), Dordrecht, Kluwer Acad. Publ. Zbl0877.35090MR1451399
  29. J. Sjöstrand, Resonances for bottles and trace formulae, Math. Nachr. 221 (2001), 95-149 Zbl0979.35109MR1806367
  30. A. V. Sobolev, Asymptotic behavior of energy levels of a quantum particle in a homogeneous magnetic field perturbed by an attenuating electric field. II, Probl. Mat. Fiz., Leningrad. Univ. 11 (1986), 232-248 MR857118
  31. X. P. Wang, Barrier resonances in strong magnetic fields, Commun. Partial Differ. Equations 17 (1992), 1539-1566 Zbl0795.35097MR1187621

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