Rate of decay at high energy of local spectral projections associated with Schrödinger operators

Kenji YAJIMA

doi:10.2969/jmsj/04110117

Kenji YAJIMA

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JOURNAL FREE ACCESS

1989 Volume 41 Issue 1 Pages 117-142

DOI https://doi.org/10.2969/jmsj/04110117

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Published: 1989 Received: August 07, 1987 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, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 2 (1975), 151-218.
2) V. I. Arnold, Small divisor problems in classical and celestial mechanics, Uspekhi Mat. Nauk, 18 (1963), 91-192.
3) K. Asada and D. Fujiwara, On some oscillatory integral transformations in L²(Rⁿ), Japan. J. Math., 4 (1978), 299-361.
4) J. Bellissard, Small divisors in quantum mechanics, in chaotic behavior in quantum systems, ed. G. Casati, Plenum, New York and London, 1985.
5) G. Gallavotti, The elements of mechanics, Springer, 1982.
6) I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. Ser. 2, 1 (1955), 253-304.
7) L. Hörmander, The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218.
8) L. Hörmander, The analysis of linear partial differential operators I-IV, Springer, 1983-1984.
9) J. Howland, Perturbation theory of dense point spectra, preprint, Univ. Virginia, 1986.
10) H. Isozaki and H. Kitada, Micro-local resolvent estimates for 2-body Schrödinger operators, J. Funct. Anal., 57 (1984), 270-300.
11) F. John, Partial differential equations, Springer, 1971.
12) H. Kumano-go, Pseudo-differential operators, MIT Press, Cambridge, 1980.
13) H. Kumano-go, Theory of pseudo-differential and Fourier integral operators and the fundamental solution of hyperbolic equations, Lecture Note, Osaka University, 1983, (in Japanese).
14) B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory, Amer. Math. Soc., Providence, 1975.
15) J. Moser, Stable and random motion in dynamical systems, Ann. of Math. Stud., 77, Princeton Univ. Press, 1973.
16) M. Reed and B. Simon, Methods of modern mathematical physics II, Fourier Analysis, selfadjointness and IV, Analysis of operators, Academic Press, New York-San Francisco-London, 1975 and 1978.
17) Y. Saito, The principle of limiting absorption for the non-selfadjoint Schrödinger operators in R^N (N=2), Publ. Res. Inst. Math. Sci., 9(1974), 397-428.
18) B. Simon, Schrödinger semi-groups, Bull. Amer. Math. Soc. (N. S.), 7 (1982), 447-526.
19) H. Tamura, Asymptotic formulas with sharp remainder estimates for eigen-values of elliptic operators of second order, Duke Math. J., 49 (1982), 87-111.
20) E. C. Titchmarsh, Eigenfunction expansions associated with second order differential equations, part one, Oxford Univ. Press, London, 1962.
21) B. R. Vainberg, On the parametrix and asymptotics of the spectral function of differential operators in Rⁿ, Soviet Math. Dokl., 31 (1985), 456-460.
22) K. Yajima and H. Kitada, Bound states and scattering states for time periodic Hamiltonians, Ann. Inst. H. Poincaré, Sec. A., 39 (1983), 145-157.
23) K. Yajima, Large time behaviors of time periodic quantum systems, in Differential Equation, ed. I. W. Knowles and R. L. Lewis, North Holland, 1984.
24) K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.

Right : [1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 2 (1975), 151-218.
[2] V. I. Arnold, Small divisor problems in classical and celestial mechanics, Uspekhi Mat. Nauk, 18 (1963), 91-192.
[3] K. Asada and D. Fujiwara, On some oscillatory integral transformations in L²(Rⁿ), Japan. J. Math., 4 (1978), 299-361.
[4] J. Bellissard, Small divisors in quantum mechanics, in chaotic behavior in quantum systems, ed. G. Casati, Plenum, New York and London, 1985.
[5] G. Gallavotti, The elements of mechanics, Springer, 1982.
[6] I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. Ser. 2, 1 (1955), 253-304.
[7] L. Hörmander, The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218.
[8] L. Hörmander, The analysis of linear partial differential operators I-IV, Springer, 1983-1984.
[9] J. Howland, Perturbation theory of dense point spectra, preprint, Univ. Virginia, 1986.
[10] H. Isozaki and H. Kitada, Micro-local resolvent estimates for 2-body Schrödinger operators, J. Funct. Anal., 57 (1984), 270-300.
[11] F. John, Partial differential equations, Springer, 1971.
[12] H. Kumano-go, Pseudo-differential operators, MIT Press, Cambridge, 1980.
[13] H. Kumano-go, Theory of pseudo-differential and Fourier integral operators and the fundamental solution of hyperbolic equations, Lecture Note, Osaka University, 1983, (in Japanese).
[14] B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory, Amer. Math. Soc., Providence, 1975.
[15] J. Moser, Stable and random motion in dynamical systems, Ann. of Math. Stud., 77, Princeton Univ. Press, 1973.
[16] M. Reed and B. Simon, Methods of modern mathematical physics II, Fourier Analysis, selfadjointness and IV, Analysis of operators, Academic Press, New York-San Francisco-London, 1975 and 1978.
[17] Y. Saito, The principle of limiting absorption for the non-selfadjoint Schrödinger operators in R^N (N=2), Publ. Res. Inst. Math. Sci., 9(1974), 397-428.
[18] B. Simon, Schrödinger semi-groups, Bull. Amer. Math. Soc. (N. S.), 7 (1982), 447-526.
[19] H. Tamura, Asymptotic formulas with sharp remainder estimates for eigen-values of elliptic operators of second order, Duke Math. J., 49 (1982), 87-111.
[20] E. C. Titchmarsh, Eigenfunction expansions associated with second order differential equations, part one, Oxford Univ. Press, London, 1962.
[21] B. R. Vainberg, On the parametrix and asymptotics of the spectral function of differential operators in Rⁿ, Soviet Math. Dokl., 31 (1985), 456-460.
[22] K. Yajima and H. Kitada, Bound states and scattering states for time periodic Hamiltonians, Ann. Inst. H. Poincaré, Sec. A., 39 (1983), 145-157.
[23] K. Yajima, Large time behaviors of time periodic quantum systems, in Differential Equation, ed. I. W. Knowles and R. L. Lewis, North Holland, 1984.
[24] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.

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