logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
Mitya Boyarchenko; Sergei Levendorski
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, 56 no. 6 (2006), p. 1827-1901, doi: 10.5802/aif.2229
Article PDF | Reviews MR 2282677 | Zbl 1127.35028
Class. Math.: 35P20, 35J10, 22E25
Keywords: Schrödinger operators, spectral asymptotics, orbit method, nilpotent Lie algebras

Résumé - Abstract

We present a pair of conjectural formulas that compute the leading term of the spectral asymptotics of a Schrödinger operator on $L^2(\mathbb{R}^n)$ 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.

Bibliography

[1] D. Arnal & J.-C. Cortet, “Répresentations $*$ des groupes exponentiels”, J. Funct. Anal. 92 (1990) no. 1, p. 103-135 Article |  MR 1064689 |  Zbl 0726.22011
[2] P. Bernat, C. Conze, M. Duflo, N. Lévy-Nahas, M. Rais, P. Renouard & M. Vzationergne, Représentations des groupes de Lie résolubles, Monographies, 4, Soc. Math. de France, 1972  Zbl 0248.22012
[3] P. Bonnet, “Paramétrisation du dual d’une algèbre de Lie nilpotente”, Ann. Inst. Fourier 38 (1988) no. 3, p. 169-197 Cedram |  MR 976688 |  Zbl 0618.22004
[4] M. Boyarchenko & S. Levendorskiĭ, “Generalizations of the classical Weyl and Colin de Verdière’s formulas and the orbit method”, Proc. Natl. Acad. Sci. USA 102 (2005) no. 16, p. 5663-5668 Article |  MR 2142891 |  Zbl pre05169142
[5] Y. Colin de Verdière, “L’asymptotique de Weyl pour les bouteilles magnétiques”, Comm. Math. Phys. 105 (1986), p. 327-335 Article |  MR 849211 |  Zbl 0612.35102
[6] H. L. Cycon, R. G. Froese, W. Kirsch & B. Simon, Schrödinger operators with applications to quantum mechanics and global geometry, Springer-Verlag, Berlin, New York, Heidelberg, London, Paris, Tokyo, 1985  Zbl 0619.47005
[7] C. L. Fefferman, “The uncertainty principle”, Bull. Amer. Math. Soc. 9 (1983), p. 129-206 Article |  MR 707957 |  Zbl 0526.35080
[8] C. Gordon, D. Webb & S. Wolpert, “One cannot hear the shape of a drum”, Bull. Amer. Math. Soc. 27 (1992), p. 134-138 Article |  MR 1136137 |  Zbl 0756.58049
[9] D. Gurarie, “Non-classical eigenvalue asymptotics for operators of Schrödinger type”, Bull. Am. Math. Soc. 15 (1986) no. 2, p. 233-237 Article |  MR 854562 |  Zbl 0628.35076
[10] B. Helffer & A. Mohamed, “Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique”, Ann. Inst. Fourier, Grenoble 38 (1988) no. 2, p. 95-112 Cedram |  MR 949012 |  Zbl 0638.47047
[11] B. Helffer & J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Math., Boston, 1985  MR 897103 |  Zbl 0568.35003
[12] L. Hörmander, The analysis of differential operators. 3, Springer-Verlag, Berlin, New York, Heidelberg, 1985  Zbl 0601.35001
[13] V. Ivriǐ, Estimate for the number of negative eigenvalues of the Schrödinger operator with intense field, in Journées Équations aux Dérivées partielles de Saint-Jean-de-Monts, Soc. Math. France, 1987 Numdam |  Zbl 0637.35063
[14] Mark Kac, “Can one hear the shape of a drum?”, Amer. Math. Monthly 73 (1966), p. 1-23 Article |  MR 201237 |  Zbl 0139.05603
[15] A. A. Kirillov, “Unitary representations of nilpotent Lie groups”, Uspehi Mat. Nauk 17 (1962) no. 4 (106), p. 57-110  MR 142001 |  Zbl 0106.25001
[16] N. V. Krylov, Introduction to the theory of diffusion processes, Translations of Mathematical Monographs, 142, American Mathematical Society, 1995  MR 1311478 |  Zbl 0844.60050
[17] S. Z. Levendorskiǐ, “Non-classical spectral asymptotics”, Russian Math. Surveys 43 (1988) no. 1, p. 123-157 Article |  Zbl 0671.35064
[18] S. Z. Levendorskiǐ, Asymptotic distribution of eigenvalues of differential operators, Dordrecht: Kluwer Academic Publishers, 1990  MR 1079317 |  Zbl 0721.35049
[19] S. Z. Levendorskiǐ, Degenerate elliptic equations, Dordrecht: Kluwer Academic Publishers, 1993  MR 1247957 |  Zbl 0786.35063
[20] S. Z. Levendorskiǐ, “Spectral properties of Schrödinger operators with irregular magnetic potentials, for a spin $\frac{1}{2}$ particle”, J. Math. Anal. Appl. 216 (1997) no. 1, p. 48-68 Article |  MR 1487252 |  Zbl 0902.35076
[21] P. Levy-Bruhl, A. Mohamed & J. Nourrigat, “Spectral theory and representations of nilpotent groups”, Bull. Amer. Math. Soc. 26 (1992) no. 2, p. 299-303 Article |  MR 1129314 |  Zbl 0749.35030
[22] D. Manchon, “Formule de Weyl pour les groupes de Lie nilpotents”, J. Reine Angew. Math. 418 (1991), p. 77-129 Article |  MR 1111202 |  Zbl 0721.22004
[23] D. Manchon, “Weyl symbolic calculus on any Lie group”, Acta Appl. Math. 30 (1993) no. 2, p. 159-186 Article |  MR 1204731 |  Zbl 0779.22005
[24] D. Manchon, “Opérateurs pseudodifférentiels et représentations unitaires des groupes de Lie”, Bull. Soc. Math. France 123 (1995) no. 1, p. 117-138 Numdam |  MR 1330790 |  Zbl 0826.22009
[25] D. Manchon, “Distributions à support compact et représentations unitaires”, J. Lie Theory 9 (1999) no. 2, p. 403-424  MR 1718231 |  Zbl 1012.22024
[26] A. Mohamed & J. Nourrigat, “Encadrement du $N(\lambda )$ pour des opérateurs de Schrödinger avec champ magnétique”, J. Math. Pures Appl. 70 (1991) no. 9, p. 87-99  MR 1091921 |  Zbl 0725.35068
[27] N. Nilsson, “Asymptotic estimates for spectral functions connected with hypoelliptic differential operators”, Ark. Mat. 5 (1965), p. 527-540 Article |  MR 218931 |  Zbl 0144.36302
[28] N. Nilsson, “Some growth and ramification properties of certain integrals on algebraic manifolds”, Ark. Mat. 5 (1965), p. 463-476 Article |  MR 175904 |  Zbl 0168.42004
[29] N. V. Pedersen, “On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications, part I”, Math. Ann. 281 (1988), p. 633-669 Article |  MR 958263 |  Zbl 0629.22004
[30] L. Pukanszky, “On the theory of exponential groups”, Trans. Amer. Math. Soc. 126 (1967), p. 487-507 Article |  MR 209403 |  Zbl 0207.33605
[31] L. Pukanszky, “Unitary representations of solvable Lie groups”, Ann. Sci. École Norm. Sup. (4) (1971) no. 4, p. 457-608 Numdam |  MR 439985 |  Zbl 0238.22010
[32] D. Robert, “Comportement asymptotique des valeurs propres d’opérateurs de type Schrödinger à potentiel “dégénéré””, J. Math. Pures Appl. 61 (1982), p. 275-300  MR 690397 |  Zbl 0511.35069
[33] G. V. Rozenbljum, “Asymptotic behavior of the eigenvalues of the Schrödinger operator”, Mat. Sb. (N.S.) 93 (1974) no. 135, p. 347-367, 487  MR 361470 |  Zbl 0304.35070
[34] G. V. Rozenbljum, M. Z. Solomyak & M. A. Shubin, Spectral theory of differential operators, Contemporary problems of mathematics, Itogi Nauki i Tekhniki VINITI, 1989  MR 1033500 |  Zbl 0715.35057
[35] B. Simon, “Nonclassical eigenvalue asymptotics”, J. Funct.Anal. 53 (1983) no. 1, p. 84-98 Article |  MR 715548 |  Zbl 0529.35064
[36] M. Z. Solomyak, “Asymptotics of the spectrum of the Schrödinger operator with non-regular homogeneous potential”, Math. USSR Sbornik 55 (1986) no. 1, p. 19-37 Article |  Zbl 0657.35099
[37] H. Tamura, “Asymptotic distribution of eigenvalues for Schrödinger operators with magnetic fields”, Nagoya Math. J. 105 (1987), p. 49-69 Article |  MR 881008 |  Zbl 0623.35048
[38] V. N. Tulovskiǐ & M. A. Shubin, “The asymptotic distribution of the eigenvalues of pseudodifferential operators in $R^n$”, Mat. Sb. (N.S.) 92 (1973) no. 134, p. 571-588, 648  MR 331131 |  Zbl 0295.35068
[39] M. Vergne, “La structure de Poisson sur l’algèbre symétrique d’une algèbre de Lie nilpotente”, Bull. SMF 100 (1972), p. 301-335 Numdam |  MR 379752 |  Zbl 0256.17002
top