Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields
Annales de l'Institut Fourier, Volume 56 (2006) no. 6, p. 1827-1901
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.
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.
DOI : https://doi.org/10.5802/aif.2229
Classification:  35P20,  35J10,  22E25
Keywords: Schrödinger operators, spectral asymptotics, orbit method, nilpotent Lie algebras
@article{AIF_2006__56_6_1827_0,
     author = {Boyarchenko, Mitya and Levendorski, Sergei},
     title = {Beyond the classical Weyl  and Colin de Verdi\`ere's formulas for Schr\"odinger operators  with polynomial magnetic and electric fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {6},
     year = {2006},
     pages = {1827-1901},
     doi = {10.5802/aif.2229},
     zbl = {1127.35028},
     mrnumber = {2282677},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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, Volume 56 (2006) no. 6, pp. 1827-1901. doi : 10.5802/aif.2229. https://aif.centre-mersenne.org/item/AIF_2006__56_6_1827_0/

[1] Arnal, D.; Cortet, J.-C. Répresentations * des groupes exponentiels, J. Funct. Anal., Tome 92 (1990) no. 1, pp. 103-135 | Article | MR 1064689 | Zbl 0726.22011

[2] Bernat, P.; Conze, C.; Duflo, M.; Lévy-Nahas, N.; Rais, M.; Renouard, P.; Vzationergne, M. Représentations des groupes de Lie résolubles, Monographies, Soc. Math. de France (4) (1972) | Zbl 0248.22012

[3] Bonnet, P. Paramétrisation du dual d’une algèbre de Lie nilpotente, Ann. Inst. Fourier, Tome 38 (1988) no. 3, pp. 169-197 | Article | Numdam | MR 976688 | Zbl 0618.22004

[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, Tome 102 (2005) no. 16, pp. 5663-5668 | Article | MR 2142891 | Zbl pre05169142

[5] Colin De Verdière, Y. L’asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., Tome 105 (1986), pp. 327-335 | Article | MR 849211 | Zbl 0612.35102

[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 0619.47005

[7] Fefferman, C. L. The uncertainty principle, Bull. Amer. Math. Soc., Tome 9 (1983), pp. 129-206 | Article | MR 707957 | Zbl 0526.35080

[8] Gordon, C.; Webb, D.; Wolpert, S. One cannot hear the shape of a drum, Bull. Amer. Math. Soc., Tome 27 (1992), pp. 134-138 | Article | MR 1136137 | Zbl 0756.58049

[9] Gurarie, D. Non-classical eigenvalue asymptotics for operators of Schrödinger type, Bull. Am. Math. Soc., Tome 15 (1986) no. 2, pp. 233-237 | Article | MR 854562 | Zbl 0628.35076

[10] Helffer, B.; Mohamed, A. Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique, Ann. Inst. Fourier, Grenoble, Tome 38 (1988) no. 2, pp. 95-112 | Article | Numdam | MR 949012 | Zbl 0638.47047

[11] Helffer, B.; Nourrigat, J. Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Math., Boston (1985) | MR 897103 | Zbl 0568.35003

[12] Hörmander, L. The analysis of differential operators. 3, Springer-Verlag, Berlin, New York, Heidelberg (1985) | Zbl 0601.35001

[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 0637.35063

[14] Kac, Mark Can one hear the shape of a drum?, Amer. Math. Monthly, Tome 73 (1966), pp. 1-23 | Article | MR 201237 | Zbl 0139.05603

[15] Kirillov, A. A. Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk, Tome 17 (1962) no. 4 (106), pp. 57-110 | MR 142001 | Zbl 0106.25001

[16] Krylov, N. V. Introduction to the theory of diffusion processes, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI (142) (1995) | MR 1311478 | Zbl 0844.60050

[17] Levendorskiǐ, S. Z. Non-classical spectral asymptotics, Russian Math. Surveys, Tome 43 (1988) no. 1, pp. 123-157 | Article | Zbl 0671.35064

[18] Levendorskiǐ, S. Z. Asymptotic distribution of eigenvalues of differential operators, Dordrecht: Kluwer Academic Publishers (1990) | MR 1079317 | Zbl 0721.35049

[19] Levendorskiǐ, S. Z. Degenerate elliptic equations, Dordrecht: Kluwer Academic Publishers (1993) | MR 1247957 | Zbl 0786.35063

[20] Levendorskiǐ, S. Z. Spectral properties of Schrödinger operators with irregular magnetic potentials, for a spin 1 2 particle, J. Math. Anal. Appl., Tome 216 (1997) no. 1, pp. 48-68 | Article | MR 1487252 | Zbl 0902.35076

[21] Levy-Bruhl, P.; Mohamed, A.; Nourrigat, J. Spectral theory and representations of nilpotent groups, Bull. Amer. Math. Soc., Tome 26 (1992) no. 2, pp. 299-303 | Article | MR 1129314 | Zbl 0749.35030

[22] Manchon, D. Formule de Weyl pour les groupes de Lie nilpotents, J. Reine Angew. Math., Tome 418 (1991), pp. 77-129 | Article | MR 1111202 | Zbl 0721.22004

[23] Manchon, D. Weyl symbolic calculus on any Lie group, Acta Appl. Math., Tome 30 (1993) no. 2, pp. 159-186 | Article | MR 1204731 | Zbl 0779.22005

[24] Manchon, D. Opérateurs pseudodifférentiels et représentations unitaires des groupes de Lie, Bull. Soc. Math. France, Tome 123 (1995) no. 1, pp. 117-138 | Numdam | MR 1330790 | Zbl 0826.22009

[25] Manchon, D. Distributions à support compact et représentations unitaires, J. Lie Theory, Tome 9 (1999) no. 2, pp. 403-424 | MR 1718231 | Zbl 1012.22024

[26] Mohamed, A.; Nourrigat, J. Encadrement du N(λ) pour des opérateurs de Schrödinger avec champ magnétique, J. Math. Pures Appl., Tome 70 (1991) no. 9, pp. 87-99 | MR 1091921 | Zbl 0725.35068

[27] Nilsson, N. Asymptotic estimates for spectral functions connected with hypoelliptic differential operators, Ark. Mat., Tome 5 (1965), pp. 527-540 | Article | MR 218931 | Zbl 0144.36302

[28] Nilsson, N. Some growth and ramification properties of certain integrals on algebraic manifolds, Ark. Mat., Tome 5 (1965), pp. 463-476 | Article | MR 175904 | Zbl 0168.42004

[29] Pedersen, N. V. On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications, part I, Math. Ann., Tome 281 (1988), pp. 633-669 | Article | MR 958263 | Zbl 0629.22004

[30] Pukanszky, L. On the theory of exponential groups, Trans. Amer. Math. Soc., Tome 126 (1967), pp. 487-507 | Article | MR 209403 | Zbl 0207.33605

[31] Pukanszky, L. Unitary representations of solvable Lie groups, Ann. Sci. École Norm. Sup. (4) (1971) no. 4, pp. 457-608 | Numdam | MR 439985 | Zbl 0238.22010

[32] Robert, D. Comportement asymptotique des valeurs propres d’opérateurs de type Schrödinger à potentiel “dégénéré”, J. Math. Pures Appl., Tome 61 (1982), pp. 275-300 | MR 690397 | Zbl 0511.35069

[33] Rozenbljum, G. V. Asymptotic behavior of the eigenvalues of the Schrödinger operator, Mat. Sb. (N.S.), Tome 93 (1974) no. 135, p. 347-367, 487 | MR 361470 | Zbl 0304.35070

[34] Rozenbljum, G. V.; Solomyak, M. Z.; Shubin, M. A. Spectral theory of differential operators, Contemporary problems of mathematics, Itogi Nauki i Tekhniki VINITI, Moscow: VINITI, Tome 64 (1989) | MR 1033500 | Zbl 0715.35057

[35] Simon, B. Nonclassical eigenvalue asymptotics, J. Funct.Anal., Tome 53 (1983) no. 1, pp. 84-98 | Article | MR 715548 | Zbl 0529.35064

[36] Solomyak, M. Z. Asymptotics of the spectrum of the Schrödinger operator with non-regular homogeneous potential, Math. USSR Sbornik, Tome 55 (1986) no. 1, pp. 19-37 | Article | Zbl 0657.35099

[37] Tamura, H. Asymptotic distribution of eigenvalues for Schrödinger operators with magnetic fields, Nagoya Math. J., Tome 105 (1987), pp. 49-69 | MR 881008 | Zbl 0623.35048

[38] Tulovskiǐ, V. N.; Shubin, M. A. The asymptotic distribution of the eigenvalues of pseudodifferential operators in R n , Mat. Sb. (N.S.), Tome 92 (1973) no. 134, p. 571-588, 648 | MR 331131 | Zbl 0295.35068

[39] Vergne, M. La structure de Poisson sur l’algèbre symétrique d’une algèbre de Lie nilpotente, Bull. SMF, Tome 100 (1972), pp. 301-335 | Numdam | MR 379752 | Zbl 0256.17002