×

zbMATH — the first resource for mathematics

Semiclassical analysis of low lying eigenvalues. I: Non-degenerate minima: Asymptotic expansions. (English) Zbl 0526.35027

MSC:
35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
81Q15 Perturbation theories for operators and differential equations in quantum theory
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] R. Ahlrichs , Convergence Properties of the Intermolecular Force Series (1/r-Expansion) , Theo. Chim. Acta , t. 41 , 1976 , p. 7 .
[2] J. Avron , I. Herbst and B. Simon , Schrödinger Operators in Magnetic Fields III. Atoms and Ions in Constant Fields , Commun. Math. Phys. , t. 79 , 1981 , p. 529 - 572 . Article | MR 623966 | Zbl 0464.35086 · Zbl 0464.35086 · doi:10.1007/BF01209311 · minidml.mathdoc.fr
[3] J.M. Combes , P. Duclos and R. Seiler , Krein’s Formula and One Dimensional Multiple Wells , J. Func. Anal. , to appear. Zbl 0562.47002 · Zbl 0562.47002 · doi:10.1016/0022-1236(83)90085-X
[4] J.M. Combes and R. Seiler , Regularity and Asymptotic Properties of the Discrete Spectrum of Electronic Hamiltonians , Int. J. Quant. Chem. , t. 14 , 1978 , p. 213 .
[5] J. Combes and L. Thomas , Asymptotic Behavior of Eigenfunctions for Multiparticle Schrödinger Operators , Commun. Math. Phys. , t. 34 , 1973 , p. 251 - 270 . Article | MR 391792 | Zbl 0271.35062 · Zbl 0271.35062 · doi:10.1007/BF01646473 · minidml.mathdoc.fr
[6] I. Herbst and B. Simon , Dilation Analyticity in Constant Electric Field, II. The N-Body Problem, Borel Summability , Commun. Math. Phys. , t. 80 , 1981 , p. 181 - 216 . Article | MR 623157 | Zbl 0473.47038 · Zbl 0473.47038 · doi:10.1007/BF01213010 · minidml.mathdoc.fr
[7] W. Hunziker and C. Pillet , Commun. Math. Phys. , to appear.
[8] R. Ismigilov , Conditions for the Semiboundedness and Discreteness of the Spectrum for One-Dimensional Differential Equations , Soviet Math. Dokl. , t. 2 , 1961 , p. 1137 . Zbl 0286.34031 · Zbl 0286.34031
[9] T. Kato , Perturbation Theory for Linear Operators , Springer , 1966 . Zbl 0148.12601 · Zbl 0148.12601
[10] R. Marcus , D.W. Noid and M.L. Koszykowski , Semiclassical Studies of Bound States and Molecular Dynamics , Springer Lecture Notes in Physics , t. 91 , 1978 , p. 283 . MR 550902
[11] W. Miller , Classical Limit Quantum Mechanics and the Theory of Molecular Collisions , Adv. Chem. Phys. , t. 25 , 1974 , p. 69 .
[12] J. Morgan , Schrödinger Operators Whose Potentials Have Separated Singularities , J. Op. Th. , t. 1 , 1979 , p. 1 . MR 526292 | Zbl 0439.35022 · Zbl 0439.35022
[13] J. Morgan and B. Simon , On the Asymptotics of Born Oppenheimer Curves for Large Nuclear Separations , Int. J. Quant. Chem. , t. 17 , 1980 , p. 1143 - 1166 .
[14] M. Reed and B. Simon , Methods of Modern Mathematical Physics, IV. Analvsis of Operators , Academic Press , 1978 . MR 493421 | Zbl 0401.47001 · Zbl 0401.47001
[15] I. Sigal , Geometric Parametrices in the QM N-Body Problem , Duke Math. J. , to appear. MR 705038
[16] I. Sigal , Geometric Methods in the Quantum Many Body Problem, Nonexistence of Very Negative Ions , Commun. Math. Phys. , t. 85 , 1982 , p. 309 - 324 . Article | MR 676004 | Zbl 0503.47041 · Zbl 0503.47041 · doi:10.1007/BF01254462 · minidml.mathdoc.fr
[17] B. Simon , Coupling Constant Analyticity for the Anharmonic Oscillator (with an appendix by A. Dicke) , Ann. Phys. , t. 58 , 1970 , p. 76 - 136 . MR 416322
[18] B. Simon , Spectrum and Continuum Eigenfunctions of Schrödinger Operators , J. Func. Anal. , t. 42 , 1981 , p. 347 - 355 . MR 626449 | Zbl 0471.47028 · Zbl 0471.47028 · doi:10.1016/0022-1236(81)90094-X
[19] B. Simon , Schrödinger Semigroups , Bull. Am. Math. Soc. , t. 1 , 1982 , p. 447 - 526 . Article | MR 670130 | Zbl 0524.35002 · Zbl 0524.35002 · doi:10.1090/S0273-0979-1982-15041-8 · minidml.mathdoc.fr
[20] B. Simon , Semiclassical Analysis of Low Lying Eigenvalues, II. Tunneling , in prep. Zbl 0626.35070 · Zbl 0626.35070 · doi:10.2307/2007072
[21] E. Witten , Supersymmetry and Morse Theory , Princeton Preprint. MR 683171 · Zbl 0499.53056
[22] E.B. Davies , The Twisting Trick for Double Well Hamiltonians , Commun. Math. Phys. , t. 85 , 1982 , p. 471 - 479 . Article | MR 678157 | Zbl 0524.47019 · Zbl 0524.47019 · doi:10.1007/BF01208725 · minidml.mathdoc.fr
[23] Additional earlier papers on the one dimensional case include: (a) J.M. Combes , Seminar on Spectral and Scattering Theory (ed. S. Kuroda), RIMS Publication 242 , 1975 , p. 22 - 38 . (b) J.M. Combes , The Born Oppenheimer Approximation , in The Schrödinger Equation (ed. W. Thirring and P. Urban), Springer , 1976 , p. 22 - 38 . (c) J.M. Combes and R. Seiler , in Quantum Dynamics of Molecules (ed. G. Wooley), Plenum , 1980 . (d) J.M. Combes , P. Duclos and R. Seiler , in Rigorous Atomic and Molecular Physics (ed. G. Velo and A. Wightman), Plenum , 1981 .
[24] A sketch of Reference 20 appears in B. Simon , Instantons, Double Wells and Large Deviations , Bull. AMS , March, 1983 issue. Article | Zbl 0529.35059 · Zbl 0529.35059 · doi:10.1090/S0273-0979-1983-15104-2 · minidml.mathdoc.fr
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.