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The asymptotic formulas for the number of bound states in the strong coupling limit
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1984 Volume 36 Issue 3 Pages 355-374
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- Published: 1984 Received: October 04, 1982 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) S. Agmon, On kernels, eigenvalues and eigenfunctions of operators related to elliptic problems, Comm. Pure Appl. Math., 18 (1965), 627-663.
2) S. Agmon, Asymptotic formulas with remainder estimates for eigenvalues of elliptic operators, Israel J. Math., 5 (1968), 165-183.
3) R. Beals, A general calculus of pseudodifferential operators, Duke Math. J., 42 (1975), 1-42.
4) M. Sh. Birman, On the spectrum of singular boundary problems, Mat. Sb., 55 (1961), 125-174; English transl., Amer. Math. Soc. Transl., 53 (1966), 23-80.
5) M. Sh. Birman and V. V. Borzov, On the asymptotics of the discrete spectrum of some singular operators, Prob. Math. Phys., 5 (1971), 24-37, Leninglard; English Transl., Topics Math. Phys., 5 (1972), 19-30, Plenum Press.
6) Colin de Verdière, Spectre conjoint d'opérateurs pseudodifférentiels qui commutent I, Duke Math. J., 46 (1979), 169-182.
7) B. Helffer and D. Robert, Calcul fonctionnel par la transformation de Mellin et operateurs admissibles, J. Funct. Anal., 53 (1983), 246-268.
8) L. Hörmander, The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218.
9) V. Ya. Ivrii, On the second term of the spectral asymptotics for the Laplace-Beltrami operator on manifold with boundary, Funkts. Anal. i Pril., 14, no. 2 (1980), 25-34; English transl., Funct. Anal. Appl., 14, 99-106.
10) M. Kac, On the asymptotic number of bound states for certain attractive potentials, Topics Funct. Anal., Adv. in Math. Suppl. Studies, 3 (1978), 159-167.
11) H. Kumano-go, A calculus of Fourier integral operator on Rn and the fundamental solution for an operator of hyperbolic type, Comm. Partial Differential Equations, 1 (1976), 1-44.
12) H. Kumano-go, Pseudo-differential operators, MIT Press, 1980.
13) E. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem, Proc. Sympos. Pure Math., Amer. Math. Soc., 36 (1980), 241-252.
14) A. Martin, Bound states in the strong coupling limit, Helv. Phys. Acta, 45 (1972), 140-148.
15) R. B. Merlose, Weyl's conjecture for manifolds with concave boundary, Proc. Sympos. Pure Math., Amer. Math. Soc., 36 (1980), 257-274.
16) A. Pleijel, On a theorem of P. Malliavin, Israel J. Math., 1 (1963), 166-168.
17) M. Reed and B. Simon, Analysis of operators, Academic Press, 1979.
18) D. Robert, Propriétés spectrales d'opérateurs pseudo-différentiels, Comm. Partial Differential Equations, 3 (1978), 755-826.
19) G. V. Rosenbljum, The distribution of the discrete spectrum of singular differential operators, Dokl. Akad. Nauk SSSR, 202 (1972), 1012-1015; English transl., Soviet Math. Dokl., 13 (1972), 245-249.
20) J. Schwinger, On the bound states of a given potential, Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 122-129.
21) R. T. Seeley, A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of R3, Adv. in Math., 29 (1978), 244-269.
22) H. Tamura, Asymptotic formulas with sharp remainder estimates for bound states of Schrödinger operators, I, J. Analyse Math., 40 (1981), 166-182.
23) H. Tamura, Asymptotic formulas with sharp remainder estimates for bound states of Schrödinger operators, II, J. Analyse Math., 41 (1982), 85-108.
24) F. Treves, Introduction to pseudodifferential and Fourier integral operators, vols. 1 and 2, Plenum Press, 1980.
Right : [1] S. Agmon, On kernels, eigenvalues and eigenfunctions of operators related to elliptic problems, Comm. Pure Appl. Math., 18 (1965), 627-663.
[2] S. Agmon, Asymptotic formulas with remainder estimates for eigenvalues of elliptic operators, Israel J. Math., 5 (1968), 165-183.
[3] R. Beals, A general calculus of pseudodifferential operators, Duke Math. J., 42 (1975), 1-42.
[4] M. Sh. Birman, On the spectrum of singular boundary problems, Mat. Sb., 55 (1961), 125-174; English transl., Amer. Math. Soc. Transl., 53 (1966), 23-80.
[5] M. Sh. Birman and V. V. Borzov, On the asymptotics of the discrete spectrum of some singular operators, Prob. Math. Phys., 5 (1971), 24-37, Leninglard; English Transl., Topics Math. Phys., 5 (1972), 19-30, Plenum Press.
[6] Colin de Verdière, Spectre conjoint d'opérateurs pseudodifférentiels qui commutent I, Duke Math. J., 46 (1979), 169-182.
[7] B. Helffer and D. Robert, Calcul fonctionnel par la transformation de Mellin et operateurs admissibles, J. Funct. Anal., 53 (1983), 246-268.
[8] L. Hörmander, The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218.
[9] V. Ya. Ivrii, On the second term of the spectral asymptotics for the Laplace-Beltrami operator on manifold with boundary, Funkts. Anal. i Pril., 14, no. 2 (1980), 25-34; English transl., Funct. Anal. Appl., 14, 99-106.
[10] M. Kac, On the asymptotic number of bound states for certain attractive potentials, Topics Funct. Anal., Adv. in Math. Suppl. Studies, 3 (1978), 159-167.
[11] H. Kumano-go, A calculus of Fourier integral operator on Rn and the fundamental solution for an operator of hyperbolic type, Comm. Partial Differential Equations, 1 (1976), 1-44.
[12] H. Kumano-go, Pseudo-differential operators, MIT Press, 1980.
[13] E. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem, Proc. Sympos. Pure Math., Amer. Math. Soc., 36 (1980), 241-252.
[14] A. Martin, Bound states in the strong coupling limit, Helv. Phys. Acta, 45 (1972), 140-148.
[15] R. B. Merlose, Weyl's conjecture for manifolds with concave boundary, Proc. Sympos. Pure Math., Amer. Math. Soc., 36 (1980), 257-274.
[16] A. Pleijel, On a theorem of P. Malliavin, Israel J. Math., 1 (1963), 166-168.
[17] M. Reed and B. Simon, Analysis of operators, Academic Press, 1979.
[18] D. Robert, Propriétés spectrales d'opérateurs pseudo-différentiels, Comm. Partial Differential Equations, 3 (1978), 755-826.
[19] G. V. Rosenbljum, The distribution of the discrete spectrum of singular differential operators, Dokl. Akad. Nauk SSSR, 202 (1972), 1012-1015; English transl., Soviet Math. Dokl., 13 (1972), 245-249.
[20] J. Schwinger, On the bound states of a given potential, Proc. Nat. Acad. Sci. U. S. A., 47 (1961), 122-129.
[21] R. T. Seeley, A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of R3, Adv. in Math., 29 (1978), 244-269.
[22] H. Tamura, Asymptotic formulas with sharp remainder estimates for bound states of Schrödinger operators, I, J. Analyse Math., 40 (1981), 166-182.
[23] H. Tamura, Asymptotic formulas with sharp remainder estimates for bound states of Schrödinger operators, II, J. Analyse Math., 41 (1982), 85-108.
[24] F. Treves, Introduction to pseudodifferential and Fourier integral operators, vols. 1 and 2, Plenum Press, 1980.
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