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

Shen, Zhongwei

doi:10.1090/S0002-9947-96-01709-6

Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields
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by Zhongwei Shen PDF

Trans. Amer. Math. Soc. 348 (1996), 4465-4488 Request permission

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

Shmuel Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$-body Schrödinger operators, Mathematical Notes, vol. 29, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982. MR 745286
J. Avron, I. Herbst, and B. Simon, Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J. 45 (1978), no. 4, 847–883. MR 518109, DOI 10.1215/S0012-7094-78-04540-4
Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
David Gurarie, Nonclassical eigenvalue asymptotics for operators of Schrödinger type, Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 233–237. MR 854562, DOI 10.1090/S0273-0979-1986-15488-1
Bernard Helffer, Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, vol. 1336, Springer-Verlag, Berlin, 1988. MR 960278, DOI 10.1007/BFb0078115
B. Helffer and A. Mohamed, Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 2, 95–112 (French, with English summary). MR 949012, DOI 10.5802/aif.1136
Bernard Helffer and Jean Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Mathematics, vol. 58, Birkhäuser Boston, Inc., Boston, MA, 1985 (French). MR 897103
B. Helffer and J. Nourrigat, Décroissance à l’infini des fonctions propres de l’opérateur de Schrödinger avec champ électromagnétique polynomial, J. Anal. Math. 58 (1992), 263–275 (French). Festschrift on the occasion of the 70th birthday of Shmuel Agmon. MR 1226946, DOI 10.1007/BF02790367
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
Akira Iwatsuka, Magnetic Schrödinger operators with compact resolvent, J. Math. Kyoto Univ. 26 (1986), no. 3, 357–374. MR 857223, DOI 10.1215/kjm/1250520872
A. Mohamed, P. Lévy-Bruhl, and J. Nourrigat, Étude spectrale d’opérateurs liés à des représentations de groupes nilpotents, J. Funct. Anal. 113 (1993), no. 1, 65–93 (French, with French summary). MR 1214898, DOI 10.1006/jfan.1993.1047
Herbert Leinfelder and Christian G. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981), no. 1, 1–19. MR 606167, DOI 10.1007/BF01258900
Hiroyuki Matsumoto, Classical and nonclassical eigenvalue asymptotics for magnetic Schrödinger operators, J. Funct. Anal. 95 (1991), no. 2, 460–482. MR 1092136, DOI 10.1016/0022-1236(91)90039-8
A. Mohamed and J. Nourrigat, Encadrement du $N(\lambda )$ pour un opérateur de Schrödinger avec un champ magnétique et un potentiel électrique, J. Math. Pures Appl. (9) 70 (1991), no. 1, 87–99 (French). MR 1091921
Abdérémane Mohamed and George D. Raĭkov, On the spectral theory of the Schrödinger operator with electromagnetic potential, Pseudo-differential calculus and mathematical physics, Math. Top., vol. 5, Akademie Verlag, Berlin, 1994, pp. 298–390. MR 1287671
Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
Didier Robert, Comportement asymptotique des valeurs propres d’opérateurs du type Schrödinger à potentiel “dégénéré”, J. Math. Pures Appl. (9) 61 (1982), no. 3, 275–300 (1983) (French, with English summary). MR 690397
Z. Shen, $L^{p}$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier 45(2) (1995), 513–546.
—, On the eigenvalue asymptotics of Schrödinger operators, preprint, 1995.
Barry Simon, Functional integration and quantum physics, Pure and Applied Mathematics, vol. 86, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 544188
Barry Simon, Nonclassical eigenvalue asymptotics, J. Funct. Anal. 53 (1983), no. 1, 84–98. MR 715548, DOI 10.1016/0022-1236(83)90047-2
M. Solomjak, Spectral asymptotics of Schrödinger operators with non–regular homogeneous potential, Math. USSR Sb. 55 (1986), 19–38.
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Kazuya Tachizawa, Asymptotic distribution of eigenvalues of Schrödinger operators with nonclassical potentials, Tohoku Math. J. (2) 42 (1990), no. 3, 381–406. MR 1066668, DOI 10.2748/tmj/1178227617