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Simon, Barry

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

Ann. Inst. Henri Poincaré, Sect. A 38, 295-308 (1983).
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Cited in 6 Reviews
Cited in 131 Documents

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

Keywords:

semiclassical analysis; low lying eigenvalues; non-degenerate minima; asymptotic expansions; Schrödinger operators with multiple wells; quadratic approximation
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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
[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
[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
[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
[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
[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
[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
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