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

Simon, Barry

Annales de l'I.H.P. Physique théorique, Tome 38 (1983) no. 3, pp. 295-308.

[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. | MR | Zbl

[3] J.M. Combes, P. Duclos and R. Seiler, Krein's Formula and One Dimensional Multiple Wells, J. Func. Anal., to appear. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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

[9] T. Kato, Perturbation Theory for Linear Operators, Springer, 1966. | Zbl

[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

[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 | Zbl

[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 | Zbl

[15] I. Sigal, Geometric Parametrices in the QM N-Body Problem, Duke Math. J., to appear. | MR

[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. | MR | Zbl

[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

[18] B. Simon, Spectrum and Continuum Eigenfunctions of Schrödinger Operators, J. Func. Anal., t. 42, 1981, p. 347-355. | MR | Zbl

[19] B. Simon, Schrödinger Semigroups, Bull. Am. Math. Soc., t. 1, 1982, p. 447-526. | MR | Zbl

[20] B. Simon, Semiclassical Analysis of Low Lying Eigenvalues, II. Tunneling, in prep. | Zbl

[21] E. Witten, Supersymmetry and Morse Theory, Princeton Preprint. | MR

[22] E.B. Davies, The Twisting Trick for Double Well Hamiltonians, Commun. Math. Phys., t. 85, 1982, p. 471-479. | MR | Zbl

(1) 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.

(2) A sketch of Reference 20 appears in B. Simon, Instantons, Double Wells and Large Deviations, Bull. AMS, March, 1983 issue. | Zbl