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Infinite systems of potential wells. (English) Zbl 0808.35086

This article is a short presentation of the results obtained in the last twelve years in the semiclassical study of the tunneling effect for the Schrödinger operator. The emphasis is on results where near a given energy the energy surface has an infinite number of connected components. The author presents recent results obtained by U. Carlsson [Asymptotic Anal. 3, No. 3, 189-214 (1990; Zbl 0727.35094)] concerning a general reduction for the study of the spectrum at the bottom and by F. Klopp [Ann. Inst. Henri Poincaré, Phys. Théor. 55, No. 1, 459-509 (1991; Zbl 0754.35100)] who studies the perturbations of a periodic potential.
Reviewer: B.Helffer (Paris)

MSC:

35P15 Estimates of eigenvalues in context of PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35Q40 PDEs in connection with quantum mechanics
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[1] [A] S. Agmon,Lectures on exponential decay of second order elliptic equations, Princeton Univ. Press 29(1982). · Zbl 0503.35001
[2] Briet, P.; Combes, J. M.; Duclos, P., Spectral stability under tunneling, Comm. Math. Phys., 126, 2, 249-262 (1989) · Zbl 0702.35189 · doi:10.1007/BF02125125
[3] Carlsson, U., An infinite number of wells in the semi-classical limit, J. Asymptotic Analysis, 3, 3, 189-214 (1990) · Zbl 0727.35094
[4] Combes, J. M.; Duclos, P.; Seiler, R., Krein’s formula and one dimensional multiple well, J. Funct. Anal., 52, 257-301 (1983) · Zbl 0562.47002 · doi:10.1016/0022-1236(83)90085-X
[5] Helffer, B.; Sjöstrand, J., Multiple wells in the semiclassical limit I, Comm. PDE, 9, 4, 337-408 (1984) · Zbl 0546.35053 · doi:10.1080/03605308408820335
[6] Helffer, B.; Sjöstrand, J., Puits multiples en limite semiclassique II.—Interaction moléculaire-Symétries-Perturbation, Ann. Inst. H. Poincaré Phys. Th., 42, 2, 127-212 (1985) · Zbl 0595.35031
[7] Klaus, M.; Simon, B., Coupling constant thresholds in non-relativistic quantum mechanics, Ann. of Phys., 130, 251-281 (1980) · Zbl 0455.35112 · doi:10.1016/0003-4916(80)90338-3
[8] Klopp, F., Etude semi-classique d’ une perturbation d’ un opérateur de Schrödinger périodique, Ann. Inst. Poincaré (physique théorique), 55, 1, 459-509 (1991) · Zbl 0754.35100
[9] Lithner, L., A theorem of Phragmén-Lindelöf type for second-order elliptic operators, Ark. Mat., 5, 18, 281-285 (1963) · Zbl 0161.30901
[10] Outassourt, A., Comportement semi-classique pour l’opérateur de Schrödinger à potentiel périodique, J. Funct. Anal., 72, 65-93 (1987) · Zbl 0662.35023 · doi:10.1016/0022-1236(87)90082-6
[11] Simon, B., Semi-classical analysis of low lying eigenvalues II. Tunneling, Ann. of Math., 120, 89-118 (1984) · Zbl 0626.35070 · doi:10.2307/2007072
[12] Simon, B., Semi-classical analysis of low lying eigenvalues III. Width of the ground state in strongly coupled solids, Ann. of Phys., 158, 2, 415-420 (1984) · Zbl 0596.35028 · doi:10.1016/0003-4916(84)90125-8
[13] Simon, B., The bound state of a weakly coupled Schrödinger operator in one or two dimensions, Ann. of Phys., 97, 279-288 (1976) · Zbl 0325.35029 · doi:10.1016/0003-4916(76)90038-5
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