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Comparison theorems for the gap of Schrödinger operators. (English) Zbl 0661.35062
The authors try to estimate the gap \(E_ 1-E_ 0\) for the two lowest eigenvalues, in general dimension, of the Schrödinger operator \(-\Delta +V\) under some conditions on V. The basic result is a comparison theorem for the gaps of two Schrödinger operators, with applications to explicit lower bounds of the gap.
Reviewer: C.Zuily

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35Q99 Partial differential equations of mathematical physics and other areas of application
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[1] {\scS. Agmon}, Bounds on exponential decay of eigenfunctions of Schrödinger operators, in “Schrödinger Operators” (S. Graffi, Ed.), Lecture Notes in Mathematics, Vol. 1159, Springer-Verlag, New York/Berlin.
[2] Buerling, A; Deny, J, Espaces de Dirichlet. I. le cas elementaire, Acta math., 99, 203-224, (1958) · Zbl 0089.08106
[3] Davies, E.B; Simon, B, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. funct. anal., 59, 335-395, (1984) · Zbl 0568.47034
[4] Durrett, R, Brownian motion and martingales in analysis, (1984), Wadsworth Belmont, CA · Zbl 0554.60075
[5] {\scM. Fukushima}, On the generation of Markov processes by symmetric forms, in “Proceedings, Second Japan-USSR Symposium on Probability Theory”, Lecture Notes in Mathematics, Vol. 336, Springer-Verlag, New York/Berlin.
[6] Gross, L, Logarithmic Sobolev inequalities, Amer. J. math., 97, 1061-1083, (1976) · Zbl 0318.46049
[7] Harrell, E, On the rate of asymptotic eigenvalue degeneracy, Comm. math. phys., 60, 73-95, (1978) · Zbl 0395.34023
[8] Harrell, E, Double wells, Comm. math. phys., 75, 239-261, (1980) · Zbl 0445.35036
[9] Helffer, B; Sjöstrand, J, Multiple wells in the semi-classical limit, I, Comm. partial differential equations, 9, 337-408, (1984) · Zbl 0546.35053
[10] Kac, M; Thompson, C, Phase transitions and eigenvalue degeneracy of a one-dimensional anharmonic oscillator, Stud. appl. math., 48, 257-264, (1969)
[11] Kirsch, W; Simon, B, Universal lower bounds on eigenvalue splitting for one-dimensional Schrödinger opertors, Comm. math. phys., 97, 453-460, (1985) · Zbl 0579.34014
[12] Kirsch, W; Simon, B, Lifshitz tails for periodic plus random potentials, J. statist. phys., 42, 799-808, (1986) · Zbl 0629.60077
[13] Reed, M; Simon, B, Methods of modern mathematical physics. N. analysis of operators, (1978), Academic Press New York
[14] Simon, B, Functional integration and quantum physics, (1979), Academic Press New York · Zbl 0434.28013
[15] Simon, B, Schrödinger semigroups, Bull. amer. math. soc., 7, 447-526, (1982) · Zbl 0524.35002
[16] Simon, B, Semiclassical analysis of low lying eigenvalues. II. tunneling, Ann. of math., 120, 89-118, (1984) · Zbl 0626.35070
[17] Simon, B, Semiclassical analysis of low lying eigenvalues. III. width of the ground state band in strongly coupled solids, Ann. phys. (N.Y.), 158, 415-420, (1984) · Zbl 0596.35028
[18] Thomas, L, Time dependent approach to scattering from impurities in a crystal, Comm. math. phys., 33, 335-343, (1973)
[19] {\scB. Wong, S. S. T. Yau, and S. T. Yau}, An estimate of the gap of the first two eigenvalues in the Schrödinger operator, preprint. · Zbl 0603.35070
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