Eigenvalue asymptotics and exponential decay of eigenfunctions for schrödinger operators with magnetic fields

Transactions of the American Mathematical Society

Volumn 348, Issue 11, 1996, Pages 4465-4488

  Shen, Zhongwei ^a  

^a UNIVERSITY OF KENTUCKY   (United States)

Author keywords

Eigenvalue asymptotics; Reverse h lder class; Schr dinger operator

Indexed keywords

EID: 21444460044     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-96-01709-6     Document Type: Article

References (25)

1. [A] S. Agmon, Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations, Princeton Univ. Press, 1982. MR 85f: 35019
2. [AHS] J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields. I. General Interactions, Duke Math. J. 45(4) (1978), 847-883. MR 80k: 35054
3. [F] C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129-206. MR 85f: 35001
4. p-integrability of the partial derivatives of a quasi-conformal mapping, Acta Math. 130 (1973), 265-277. MR 53: 5861
5. [Gu] D. Gurarie, Nonclassical eigenvalue asymptotics for operators of Schrödinger type, Bull. Amer. Math. Soc. 15(2) (1986), 233-237. MR 88a: 35177
6. [H] B. Helffer, Semi-Classical Analysis for the Schrödinger Operator and Applications, Lectures Notes in Math., vol. 1336, Springer-Verlag, 1988. MR 90c: 81043
7. [HMo] B. HelfFer and A. Mohamed, Caractérisation du spectre essentiel de l'opérateur de Schrödinger avec un champ magnétique, Ann. Inst. Fourier 38 (1988), 95-112. MR 90d: 35215
8. [HN1] B. Helffer and J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Math. 58, Birkhäuser, Boston, 1985. MR 88i: 35029
9. [HN2] _, Decrossance a l'infini des fonctions propres de l'opérateur de Schrödinger avec champ electromagnétique polynomial, J. Anal. Math. 58 (1992), 263-275. MR 95e: 35049
10. [Ho] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, 1983. MR 85g: 35002a
11. [I] A. Iwatsuka, Magnetic Schrödinger operators with compact resolvent, J. Math. Kyoto Univ. 26-3 (1986), 357-374. MR 87j: 35287
12. [LMN] P. Lévy-Bruhl, A. Mohamed and J. Nourrigat, Etude spectrale d'opérateurs liés à des représentations de groupes nilpotents, J. Funct. Anal. 113 (1993), 65-93. MR 94k: 22014
13. [LS] H. Leinfelder and C. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981), 1-19. MR 82d: 35073
14. [Ma] H. Matsumoto, Classical and non-classical eigenvalue asymptotics for magnetic Schrödinger operators, J. Funct. Anal. 95 (1991), 460-482. MR 92b: 35114
15. [MoN] A. Mohamed and J. Nourrigat, Encadrement du N(λ) pour un operator de Schrödinger avec un champ magnétique et un potentiel électrique, J. Math. Pures Appl. 70 (1991), 87-99. MR 92a: 35122
16. [MoR] A. Mohamed and G. Raikov, On the spectral theory of the Schrödinger operator with electromagnetic potential, Pseudo-Differential Calculus and Mathematical Physics, Akademie Verlag, 1994, pp. 298-390. MR 96e: 35122
17. [Mu] B. Muckenhoupt, Weighted norm inequality for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 45: 2461
18. [R] D. Robert, Comportement asymptotique des valeurs propres d'opérateurs du type Schrödinger à potentiel "dégénéré", J. Math. Pures Appl. 61(9) (1982), 275-300. MR 84d: 35117
19. p estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier 45(2) (1995), 513-546. CMP 95:16
20. [Sh2] _, On the eigenvalue asymptotics of Schrödinger operators, preprint, 1995.
21. [Si1] B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979. MR 84m: 81066
22. [Si2] _, Nonclassical eigenvalue asymptotics, J. Funct. Anal. 53 (1983), 84-98. MR 85i: 35113
23. [So] M. Solomjak, Spectral asymptotics of Schrödinger operators with non-regular homogeneous potential, Math. USSR Sb. 55 (1986), 19-38.
24. [St] E. Stein, Harmonic Analysis: Real-Variable Method, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993. MR 95c: 42002
25. [T] K. Tachizawa, Asymptotics distribution of eigenvalues of Schrödinger operators with nonclassical potentials, Tôhoku Math. J. 42 (1990), 381-406. MR 92a: 35125