Resonances and Spectral Shift Function near the Landau levels
Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 629-671.

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 , and define the resonances of H as the poles of this meromorphic extension. We study their distribution near any fixed Landau level 2bq, q. 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.

On étudie l’opérateur de Schrödinger magnétique en dimension 3, H=H 0 +VH 0 =(-i-A) 2 -b, A est un potentiel magnétique générant un champ magnétique constant de force b>0 fixée et V est un potentiel électrique qui décroît super-exponentiellement dans la direction du champ magnétique. On montre que la résolvante de H admet un prolongement méromorphe du plan supérieur une certaine surface de Riemann et on définit les résonances de H comme les pôles de cette extension méromorphe. On étudie leur répartition près d’un niveau de Landau fixé 2bq, q. On obtient d’abord des majorations du nombre de résonances dans des petits domaines proches de 2bq. Sous des hypothses supplémentaires, on prouve des minorations du nombre de résonances qui implique la présence d’une infinité de résonances ou bien l’absence de résonances dans certains secteurs de sommet 2bq. Finalement, on montre que la fonction de décalage spectral (FDS) associée à la paire (H,H 0 ) est la somme de mesures harmoniques associées aux résonances et de la partie imaginaire d’une fonction holomorphe. Cette formule justifie l’approximation de Breit-Wigner, implique une formule de trace à la Sjöstrand et donne des informations sur les singularités de la FDS aux niveaux de Landau.

DOI: 10.5802/aif.2270
Classification: 35P25, 35J10, 47F05, 81Q10
Keywords: Magnetic Schrödinger operators, resonances, spectral shift function, Breit-Wigner approximation

Bony, Jean-François 1; Bruneau, Vincent 1; Raikov, Georgi 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)
@article{AIF_2007__57_2_629_0,
     author = {Bony, Jean-Fran\c{c}ois and Bruneau, Vincent and Raikov, Georgi},
     title = {Resonances and {Spectral} {Shift} {Function} near the {Landau} levels},
     journal = {Annales de l'Institut Fourier},
     pages = {629--671},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {2},
     year = {2007},
     doi = {10.5802/aif.2270},
     mrnumber = {2310953},
     zbl = {1129.35053},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2270/}
}
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
SP  - 629
EP  - 671
VL  - 57
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2270/
DO  - 10.5802/aif.2270
LA  - en
ID  - AIF_2007__57_2_629_0
ER  - 
%0 Journal Article
%A Bony, Jean-François
%A Bruneau, Vincent
%A Raikov, Georgi
%T Resonances and Spectral Shift Function near the Landau levels
%J Annales de l'Institut Fourier
%D 2007
%P 629-671
%V 57
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2270/
%R 10.5802/aif.2270
%G en
%F AIF_2007__57_2_629_0
Bony, Jean-François; Bruneau, Vincent; Raikov, Georgi. Resonances and Spectral Shift Function near the Landau levels. Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 629-671. doi : 10.5802/aif.2270. https://aif.centre-mersenne.org/articles/10.5802/aif.2270/

[1] Avron, J.; Herbst, I.; Simon, B. Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J., Volume 45 (1978), pp. 847-883 | DOI | MR | Zbl

[2] Barreto, A. Sá; Zworski, M. Existence of resonances in three dimensions, Commun. Math. Phys., Volume 173 (1995), pp. 401-415 | DOI | MR | Zbl

[3] Bony, J.-F.; Sjöstrand, J. Trace formula for resonances in small domains, J. Funct. Anal., Volume 184 (2001), pp. 402-418 | DOI | MR | Zbl

[4] Bouclet, J. M. Traces formulae for relatively Hilbert-Schmidt perturbations, Asymptot. Anal., Volume 32 (2002), pp. 257-291 | MR | Zbl

[5] Bouclet, J. M. Spectral distributions for long range perturbations, J. Funct. Anal., Volume 212 (2004), pp. 431-471 | DOI | MR | Zbl

[6] Bruneau, V.; Petkov, V. Meromorphic continuation of the spectral shift function, Duke Math. J., Volume 116 (2003), pp. 389-430 | DOI | MR | Zbl

[7] Bruneau, V.; Pushnitski, A.; Raikov, G. D. Spectral shift function in strong magnetic fields, Algebra i Analysis, Volume 16 (2004), pp. 207-238 English transl.: St. Petersburg Math. J. 16 (2005), p.181-209 | MR | Zbl

[8] Delande, D.; Bommier, A.; Gay, J.-C. Positive-Energy spectrum of the hydrogen atom in a magnetic field, Phys. Rev. Lett., Volume 66 (1991), pp. 141-144 | DOI

[9] Dimassi, M.; Sjöstrand, J. Spectral asymptotics in the semi-classical limit, Lecture Notes Series, London Math. Society, Volume 268, Cambridge University Press, 1999 | MR | Zbl

[10] Dimassi, M.; Zerzeri, M. A local trace formula for resonances of perturbed periodic Schrödinger operators, J. Funct. Anal., Volume 198 (2003), pp. 142-159 | DOI | MR | Zbl

[11] Fernández, C.; Raikov, G. D. On the singularities of the magnetic spectral shift function at the Landau levels, Ann. Henri Poincaré, Volume 5 (2004), pp. 381-403 | DOI | MR | Zbl

[12] Filonov, N.; Pushnitski, A. Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains, Commun. Math. Phys., Volume 264 (2006), pp. 759-772 | DOI | MR | Zbl

[13] Froese, R. Asymptotic distribution of resonances in one dimension, J. Diff. Equa., Volume 137 (1997), pp. 251-272 | DOI | MR | Zbl

[14] Froese, R.; Waxler, R. Ground state resonances of a hydrogen atom in an intense magnetic field, Rev. Math. Phys., Volume 7 (1995), pp. 311-361 | DOI | MR | Zbl

[15] Fulton, W. Algebraic Topology, A First Course, Graduate Texts in Mathematics, Springer, 1995 | MR | Zbl

[16] Gohberg, I. C.; Krein, M. G. Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Volume 18, American Math. Society, Providence, R.I., 1969 | MR | Zbl

[17] Koplienko, L. S. Trace formula for non trace-class perturbations, Sibirsk. Mat. Zh., Volume 25 (1984), pp. 62-71 English transl.: Siberian Math. J. 25 (1984), p.735-743 | MR | Zbl

[18] Koplienko, L. S. Regularized function of spectral shift for a one-dimensional Schrödinger operator with slowly decreasing potential, Sibirsk. Mat. Zh., Volume 26 (1985), p. 72-77, 62-71 English transl.: Siberian Math. J. 26 (1985), p.365–369 | MR | Zbl

[19] Krein, M. G. On perturbation determinants and a trace formula for unitary and self-adjoint operators, Dokl. Akad. Nauk SSSR, Volume 144 (1962), pp. 268-271 | MR | Zbl

[20] Landau, L. Diamagnetismus der Metalle, Z. Physik, Volume 64 (1930), pp. 629-637 | DOI

[21] Petkov, V.; Zworski, M. Semi-classical estimates on the scattering determinant, Ann. H. Poincaré, Volume 2 (2001), pp. 675-711 | DOI | MR | Zbl

[22] Raikov, G. D. Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips, Commun. PDE, Volume 15 (1990), pp. 407-434 Errata: Commun. PDE 18 (1993), 1977–1979 | DOI | MR | Zbl

[23] Raikov, G. D. Spectral shift function for Schrödinger operators in constant magnetic fields, Cubo, Volume 7 (2005), pp. 171-199 | MR | Zbl

[24] Raikov, G. D. 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, Volume 690, Springer, 2006, pp. 451-465 | MR | Zbl

[25] Raikov, G. D.; Warzel, S. Quasi-classical versus non-classical spectral asymptotics for magnetic Schödinger operators with decreasing electric potentials, Rev. Math. Phys., Volume 14 (2002), pp. 1051-1072 | DOI | MR | Zbl

[26] Sjöstrand, J. Lectures on resonances (Preprint, www.math.polytechnique.fr/~sjoestrand/)

[27] Sjöstrand, J. A trace formula for resonances and application to semi-classical Schrödinger operator, Séminaire EDP, Exposé II, École Polytechnique (1996-1997), pp. 1-17 | Numdam | MR | Zbl

[28] Sjöstrand, J. 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., Volume 490, Dordrecht, Kluwer Acad. Publ., 1997 | MR | Zbl

[29] Sjöstrand, J. Resonances for bottles and trace formulae, Math. Nachr., Volume 221 (2001), pp. 95-149 | DOI | MR | Zbl

[30] Sobolev, A. V. 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., Volume 11 (1986), pp. 232-248 | MR

[31] Wang, X. P. Barrier resonances in strong magnetic fields, Commun. Partial Differ. Equations, Volume 17 (1992), pp. 1539-1566 | DOI | MR | Zbl

Cited by Sources: