Kirsch, Werner; Simon, Barry Comparison theorems for the gap of Schrödinger operators. (English) Zbl 0661.35062 J. Funct. Anal. 75, 396-410 (1987). 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 Cited in 1 ReviewCited in 61 Documents 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 Keywords:gap; two lowest eigenvalues; Schrödinger operator; comparison; explicit lower bounds PDFBibTeX XMLCite \textit{W. Kirsch} and \textit{B. Simon}, J. Funct. Anal. 75, 396--410 (1987; Zbl 0661.35062) Full Text: DOI References: [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: Wadsworth Belmont, CA · Zbl 0554.60075 [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: Academic Press New York [14] Simon, B., Functional Integration and Quantum Physics (1979), Academic Press: 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) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.