Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima : asymptotic expansions

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1. [1] R. Ahlrichs, Convergence Properties of the Intermolecular Force Series (1/r-Expansion), Theo. Chim. Acta, t. 41, 1976, p. 7. 
2. [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. Zbl0464.35086MR623966
3. [3] J.M. Combes, P. Duclos and R. Seiler, Krein's Formula and One Dimensional Multiple Wells, J. Func. Anal., to appear. Zbl0562.47002
4. [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. [5] J. Combes and L. Thomas, Asymptotic Behavior of Eigenfunctions for Multiparticle Schrödinger Operators, Commun. Math. Phys., t. 34, 1973, p. 251-270. Zbl0271.35062MR391792
6. [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. Zbl0473.47038MR623157
7. [7] W. Hunziker and C. Pillet, Commun. Math. Phys., to appear. 
8. [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. Zbl0286.34031
9. [9] T. Kato, Perturbation Theory for Linear Operators, Springer, 1966. Zbl0148.12601
10. [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. MR550902
11. [11] W. Miller, Classical Limit Quantum Mechanics and the Theory of Molecular Collisions, Adv. Chem. Phys., t. 25, 1974, p. 69. 
12. [12] J. Morgan, Schrödinger Operators Whose Potentials Have Separated Singularities, J. Op. Th., t. 1, 1979, p. 1. Zbl0439.35022MR526292
13. [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. [14] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analvsis of Operators, Academic Press, 1978. Zbl0401.47001MR493421
15. [15] I. Sigal, Geometric Parametrices in the QM N-Body Problem, Duke Math. J., to appear. MR705038
16. [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. Zbl0503.47041MR676004
17. [17] B. Simon, Coupling Constant Analyticity for the Anharmonic Oscillator (with an appendix by A. Dicke), Ann. Phys., t. 58, 1970, p. 76-136. MR416322
18. [18] B. Simon, Spectrum and Continuum Eigenfunctions of Schrödinger Operators, J. Func. Anal., t. 42, 1981, p. 347-355. Zbl0471.47028MR626449
19. [19] B. Simon, Schrödinger Semigroups, Bull. Am. Math. Soc., t. 1, 1982, p. 447-526. Zbl0524.35002MR670130
20. [20] B. Simon, Semiclassical Analysis of Low Lying Eigenvalues, II. Tunneling, in prep. Zbl0626.35070
21. [21] E. Witten, Supersymmetry and Morse Theory, Princeton Preprint. MR683171
22. [22] E.B. Davies, The Twisting Trick for Double Well Hamiltonians, Commun. Math. Phys., t. 85, 1982, p. 471-479. Zbl0524.47019MR678157
23. (1) Additional earlier papers on the one dimensional case include: (a) J.M. Combes, Seminar on Spectral and Scattering Theory (ed. S. Kuroda), RIMS Publication242, 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. (2) A sketch of Reference 20 appears in B. Simon, Instantons, Double Wells and Large Deviations, Bull. AMS, March, 1983 issue. Zbl0529.35059

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