Available in electronic format
Available in print format		 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

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

Author(s): Zhongwei Shen
Journal: Trans. Amer. Math. Soc. 348 (1996), 4465-4488.
MSC (1991): Primary 35P20, 35J10
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We consider the Schrödinger operator with magnetic field,

\begin{equation*}H=(\frac {1}{i}\nabla -{\overset {\rightharpoonup }{a}}(x))^{2}+V(x) \ \text { in } \mathbb {R}^{n}. \end{equation*}

Assuming that $V\ge 0$ and $|\text {curl}\, \overset {\rightharpoonup }{a}|+V+1$ is locally in certain reverse Hölder class, we study the eigenvalue asymptotics and exponential decay of eigenfunctions.

References:

[A]
S. Agmon, Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations, Princeton Univ. Press, 1982. MR 85f:35019

[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

[F]
C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129--206. MR 85f:35001

[Ge]
F. Gehring, The $L^{p}$--integrability of the partial derivatives of a quasi--conformal mapping, Acta Math. 130 (1973), 265--277. MR 53:5861

[Gu]
D. Gurarie, Nonclassical eigenvalue asymptotics for operators of Schrödinger type, Bull. Amer. Math. Soc. 15(2) (1986), 233--237. MR 88a:35177

[H]
B. Helffer, Semi--Classical Analysis for the Schrödinger Operator and Applications, Lectures Notes in Math., vol. 1336, Springer--Verlag, 1988. MR 90c:81043

[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

[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

[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

[Hö]
L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer--Verlag, 1983. MR 85g:35002a

[I]
A. Iwatsuka, Magnetic Schrödinger operators with compact resolvent, J. Math. Kyoto Univ. 26--3 (1986), 357--374. MR 87j:35287

[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

[LS]
H. Leinfelder and C. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981), 1--19. MR 82d:35073

[Ma]
H. Matsumoto, Classical and non--classical eigenvalue asymptotics for magnetic Schrödinger operators, J. Funct. Anal. 95 (1991), 460--482. MR 92b:35114

[MoN]
A. Mohamed and J. Nourrigat, Encadrement du $N(\lambda )$ pour un opérator de Schrödinger avec un champ magnétique et un potentiel électrique, J. Math. Pures Appl. 70 (1991), 87--99. MR 92a:35122

[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

[Mu]
B. Muckenhoupt, Weighted norm inequality for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207--226. MR 45:2461

[R]
D. Robert, Comportement asymptotique des valeurs propres d'opérateurs du type Schrödinger á potentiel ``dégénéré'', J. Math. Pures Appl. 61(9) (1982), 275--300. MR 84d:35117

[Sh1]
Z. Shen, $L^{p}$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier 45(2) (1995), 513--546. CMP 95:16

[Sh2]
------, On the eigenvalue asymptotics of Schrödinger operators, preprint, 1995.

[Si1]
B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979. MR 84m:81066

[Si2]
------, Nonclassical eigenvalue asymptotics, J. Funct. Anal. 53 (1983), 84--98. MR 85i:35113

[So]
M. Solomjak, Spectral asymptotics of Schrödinger operators with non--regular homogeneous potential, Math. USSR Sb. 55 (1986), 19--38.

[St]
E. Stein, Harmonic Analysis: Real--Variable Method, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993. MR 95c:42002

[T]
K. Tachizawa, Asymptotics distribution of eigenvalues of Schrödinger operators with nonclassical potentials, Tôhoku Math. J. 42 (1990), 381--406. MR 92a:35125

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35P20, 35J10

Retrieve articles in all Journals with MSC (1991): 35P20, 35J10

Additional Information:

Zhongwei Shen
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: shenz@ms.uky.edu

DOI: 10.1090/S0002-9947-96-01709-6
PII: S 0002-9947(96)01709-6
Keywords: Eigenvalue asymptotics, Schrödinger operator, reverse Hölder class
Received by editor(s): May 19, 1995
Copyright of article: Copyright 1996, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google