Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields
[Au-delà des formules classiques de Weyl et de Colin de Verdière pour les opérateurs de Shrödinger avec des champs polynomiaux électriques et magnétiques.]
Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1827-1901.

Nous donnons deux formules conjecturelles pour calculer le terme dominant du comportement asymptotique du spectre d’un opérateur de Schrödinger agissant dans L 2 ( n ) avec des polynômes quasi-homogènes comme champs électriques et magnétiques. La construction se base sur la méthode des orbites de Kirillov, et s’applique donc à n’importe quelle algèbre de Lie nilpotente. Elle est liée à la géométrie des orbites coadjointes et à certaines “intégrales algébriques” étudiées par Nilsson. En utilisant la méthode de variation directe, nous démontrons que nos formules sont correctes non seulement dans le cas régulier où s’appliquent les formules de Weyl ou Colin de Verdière, mais aussi dans certains cas “irréguliers” avec différents types de dégéréscence des potentiels.

We present a pair of conjectural formulas that compute the leading term of the spectral asymptotics of a Schrödinger operator on L 2 ( n ) with quasi-homogeneous polynomial magnetic and electric fields. The construction is based on the orbit method due to Kirillov. It makes sense for any nilpotent Lie algebra and is related to the geometry of coadjoint orbits, as well as to the growth properties of certain “algebraic integrals,” studied by Nilsson. By using the direct variational method, we prove that the formulas give the correct answer not only in the “regular” cases where the classical formulas of Weyl or Colin de Verdière are applicable but in many “irregular” cases, with different types of degeneration of potentials.

DOI : 10.5802/aif.2229
Classification : 35P20, 35J10, 22E25
Keywords: Schrödinger operators, spectral asymptotics, orbit method, nilpotent Lie algebras
Mot clés : opérateurs de Schrödinger, comportement asymptotique, la méthode des orbites, algèbre de Lie nilpotente
Boyarchenko, Mitya 1 ; Levendorski, Sergei 2

1 University of Chicago Department of Mathematics Chicago, IL 60637 (USA)
2 University of Texas Department of Economics Austin, TX (USA)
@article{AIF_2006__56_6_1827_0,
     author = {Boyarchenko, Mitya and Levendorski, Sergei},
     title = {Beyond the classical {Weyl}  and {Colin} de {Verdi\`ere{\textquoteright}s} formulas for {Schr\"odinger} operators  with polynomial magnetic and electric fields},
     journal = {Annales de l'Institut Fourier},
     pages = {1827--1901},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {6},
     year = {2006},
     doi = {10.5802/aif.2229},
     zbl = {1127.35028},
     mrnumber = {2282677},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2229/}
}
TY  - JOUR
AU  - Boyarchenko, Mitya
AU  - Levendorski, Sergei
TI  - Beyond the classical Weyl  and Colin de Verdière’s formulas for Schrödinger operators  with polynomial magnetic and electric fields
JO  - Annales de l'Institut Fourier
PY  - 2006
SP  - 1827
EP  - 1901
VL  - 56
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2229/
DO  - 10.5802/aif.2229
LA  - en
ID  - AIF_2006__56_6_1827_0
ER  - 
%0 Journal Article
%A Boyarchenko, Mitya
%A Levendorski, Sergei
%T Beyond the classical Weyl  and Colin de Verdière’s formulas for Schrödinger operators  with polynomial magnetic and electric fields
%J Annales de l'Institut Fourier
%D 2006
%P 1827-1901
%V 56
%N 6
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2229/
%R 10.5802/aif.2229
%G en
%F AIF_2006__56_6_1827_0
Boyarchenko, Mitya; Levendorski, Sergei. Beyond the classical Weyl  and Colin de Verdière’s formulas for Schrödinger operators  with polynomial magnetic and electric fields. Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1827-1901. doi : 10.5802/aif.2229. http://archive.numdam.org/articles/10.5802/aif.2229/

[1] Arnal, D.; Cortet, J.-C. Répresentations * des groupes exponentiels, J. Funct. Anal., Volume 92 (1990) no. 1, pp. 103-135 | DOI | MR | Zbl

[2] Bernat, P.; Conze, C.; Duflo, M.; Lévy-Nahas, N.; Rais, M.; Renouard, P.; Vzationergne, M. Représentations des groupes de Lie résolubles, Monographies (4), Soc. Math. de France, 1972 | Zbl

[3] Bonnet, P. Paramétrisation du dual d’une algèbre de Lie nilpotente, Ann. Inst. Fourier, Volume 38 (1988) no. 3, pp. 169-197 | DOI | Numdam | MR | Zbl

[4] Boyarchenko, M.; Levendorskiĭ, S. Generalizations of the classical Weyl and Colin de Verdière’s formulas and the orbit method, Proc. Natl. Acad. Sci. USA, Volume 102 (2005) no. 16, pp. 5663-5668 | DOI | MR

[5] Colin de Verdière, Y. L’asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., Volume 105 (1986), pp. 327-335 | DOI | MR | Zbl

[6] Cycon, H. L.; Froese, R. G.; Kirsch, W.; Simon, B. Schrödinger operators with applications to quantum mechanics and global geometry, Springer-Verlag, Berlin, New York, Heidelberg, London, Paris, Tokyo, 1985 | Zbl

[7] Fefferman, C. L. The uncertainty principle, Bull. Amer. Math. Soc., Volume 9 (1983), pp. 129-206 | DOI | MR | Zbl

[8] Gordon, C.; Webb, D.; Wolpert, S. One cannot hear the shape of a drum, Bull. Amer. Math. Soc., Volume 27 (1992), pp. 134-138 | DOI | MR | Zbl

[9] Gurarie, D. Non-classical eigenvalue asymptotics for operators of Schrödinger type, Bull. Am. Math. Soc., Volume 15 (1986) no. 2, pp. 233-237 | DOI | MR | Zbl

[10] Helffer, B.; Mohamed, A. Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique, Ann. Inst. Fourier, Grenoble, Volume 38 (1988) no. 2, pp. 95-112 | DOI | Numdam | MR | Zbl

[11] Helffer, B.; Nourrigat, J. Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Math., Boston, 1985 | MR | Zbl

[12] Hörmander, L. The analysis of differential operators. 3, Springer-Verlag, Berlin, New York, Heidelberg, 1985 | Zbl

[13] Ivriǐ, V. Estimate for the number of negative eigenvalues of the Schrödinger operator with intense field, Journées Équations aux Dérivées partielles de Saint-Jean-de-Monts, Soc. Math. France (1987) | Numdam | Zbl

[14] Kac, Mark Can one hear the shape of a drum?, Amer. Math. Monthly, Volume 73 (1966), pp. 1-23 | DOI | MR | Zbl

[15] Kirillov, A. A. Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk, Volume 17 (1962) no. 4 (106), pp. 57-110 | MR | Zbl

[16] Krylov, N. V. Introduction to the theory of diffusion processes, Translations of Mathematical Monographs (142), American Mathematical Society, Providence, RI, 1995 | MR | Zbl

[17] Levendorskiǐ, S. Z. Non-classical spectral asymptotics, Russian Math. Surveys, Volume 43 (1988) no. 1, pp. 123-157 | DOI | Zbl

[18] Levendorskiǐ, S. Z. Asymptotic distribution of eigenvalues of differential operators, Dordrecht: Kluwer Academic Publishers, 1990 | MR | Zbl

[19] Levendorskiǐ, S. Z. Degenerate elliptic equations, Dordrecht: Kluwer Academic Publishers, 1993 | MR | Zbl

[20] Levendorskiǐ, S. Z. Spectral properties of Schrödinger operators with irregular magnetic potentials, for a spin 1 2 particle, J. Math. Anal. Appl., Volume 216 (1997) no. 1, pp. 48-68 | DOI | MR | Zbl

[21] Levy-Bruhl, P.; Mohamed, A.; Nourrigat, J. Spectral theory and representations of nilpotent groups, Bull. Amer. Math. Soc., Volume 26 (1992) no. 2, pp. 299-303 | DOI | MR | Zbl

[22] Manchon, D. Formule de Weyl pour les groupes de Lie nilpotents, J. Reine Angew. Math., Volume 418 (1991), pp. 77-129 | DOI | MR | Zbl

[23] Manchon, D. Weyl symbolic calculus on any Lie group, Acta Appl. Math., Volume 30 (1993) no. 2, pp. 159-186 | DOI | MR | Zbl

[24] Manchon, D. Opérateurs pseudodifférentiels et représentations unitaires des groupes de Lie, Bull. Soc. Math. France, Volume 123 (1995) no. 1, pp. 117-138 | Numdam | MR | Zbl

[25] Manchon, D. Distributions à support compact et représentations unitaires, J. Lie Theory, Volume 9 (1999) no. 2, pp. 403-424 | MR | Zbl

[26] Mohamed, A.; Nourrigat, J. Encadrement du N(λ) pour des opérateurs de Schrödinger avec champ magnétique, J. Math. Pures Appl., Volume 70 (1991) no. 9, pp. 87-99 | MR | Zbl

[27] Nilsson, N. Asymptotic estimates for spectral functions connected with hypoelliptic differential operators, Ark. Mat., Volume 5 (1965), pp. 527-540 | DOI | MR | Zbl

[28] Nilsson, N. Some growth and ramification properties of certain integrals on algebraic manifolds, Ark. Mat., Volume 5 (1965), pp. 463-476 | DOI | MR | Zbl

[29] Pedersen, N. V. On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications, part I, Math. Ann., Volume 281 (1988), pp. 633-669 | DOI | MR | Zbl

[30] Pukanszky, L. On the theory of exponential groups, Trans. Amer. Math. Soc., Volume 126 (1967), pp. 487-507 | DOI | MR | Zbl

[31] Pukanszky, L. Unitary representations of solvable Lie groups, Ann. Sci. École Norm. Sup. (4) (1971) no. 4, pp. 457-608 | Numdam | MR | Zbl

[32] Robert, D. Comportement asymptotique des valeurs propres d’opérateurs de type Schrödinger à potentiel “dégénéré”, J. Math. Pures Appl., Volume 61 (1982), pp. 275-300 | MR | Zbl

[33] Rozenbljum, G. V. Asymptotic behavior of the eigenvalues of the Schrödinger operator, Mat. Sb. (N.S.), Volume 93 (1974) no. 135, p. 347-367, 487 | MR | Zbl

[34] Rozenbljum, G. V.; Solomyak, M. Z.; Shubin, M. A. Spectral theory of differential operators, Contemporary problems of mathematics, Volume 64, Itogi Nauki i Tekhniki VINITI, Moscow: VINITI, 1989 | MR | Zbl

[35] Simon, B. Nonclassical eigenvalue asymptotics, J. Funct.Anal., Volume 53 (1983) no. 1, pp. 84-98 | DOI | MR | Zbl

[36] Solomyak, M. Z. Asymptotics of the spectrum of the Schrödinger operator with non-regular homogeneous potential, Math. USSR Sbornik, Volume 55 (1986) no. 1, pp. 19-37 | DOI | Zbl

[37] Tamura, H. Asymptotic distribution of eigenvalues for Schrödinger operators with magnetic fields, Nagoya Math. J., Volume 105 (1987), pp. 49-69 | MR | Zbl

[38] Tulovskiǐ, V. N.; Shubin, M. A. The asymptotic distribution of the eigenvalues of pseudodifferential operators in R n , Mat. Sb. (N.S.), Volume 92 (1973) no. 134, p. 571-588, 648 | MR | Zbl

[39] Vergne, M. La structure de Poisson sur l’algèbre symétrique d’une algèbre de Lie nilpotente, Bull. SMF, Volume 100 (1972), pp. 301-335 | Numdam | MR | Zbl

Cité par Sources :