The Wk, p-continuity of wave operators for Schrödinger operators

Kenji YAJIMA

doi:10.2969/jmsj/04730551

The W^{k, p}-continuity of wave operators for Schrödinger operators

Kenji YAJIMA

著者情報

ジャーナルフリー

1995 年 47 巻 3 号 p. 551-581

DOI https://doi.org/10.2969/jmsj/04730551

詳細

Published: 1995 Received: 1993/06/03 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) R. A. Adams, Sobolev Spaces, Academic Press, New York-San Francisco-London, 1975.
2) S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Ser. IV, 2 (1975), 151-218.
3) M. Beals and W. Strauss, L^p estimates for the wave equation with a potential, preprint, 1992.
4) L. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer, Berlin-Heidelberg-New York, 1976.
5) Ph. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations, 56 (1985), 310-344.
6) J. Ginibre and M. Moulin, Hilbert space approach to quantum mechanical three body problem, Ann. Inst. H. Poincaré Sec. A, 21 (1974), 97-145.
7) J. Ginibre and G. Velo, The global Cauchy problem for some non-linear Schrödinger equations, revisited, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 2 (1985), 309-327.
8) A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.
9) A. Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions, Results in L₂(R^m), m_??_5, Duke Math. J., 47 (1980), 57-80.
10) A. Jensen and S. Nakamura, Mapping properties of functions of Schrödinger operators between L^p-spaces and Besov spaces, preprint, 1992.
11) J.-L. Journe, A. Soffer and C. D. Sogge, Decay estimated for Schrödinger operators, Comm. Pure. Appl. Math., 44 (1991), 573-604.
12) T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math., 12 (1959), 403-422.
13) T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279.
14) T. Kato and S. T. Kuroda, Theory of simple scattering and eigenfunction expansions, Functional analysis and related fields, Springer-Verlag, Berlin-Heidelberg-New York, 1970, pp. 99-131.
15) T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Reviews in Math. Phys., 1 (1989), 481-496.
16) S. T. Kuroda, An introduction to scattering theory, Lecture Notes Series, 51, Aarhus University, 1978, Aarhus, Denmark.
17) S. T. Kuroda, Scattering theory for differential operators, I and II, J. Math. Soc. Japan, 25 (1972), 75-104 and 222-234.
18) A. Melin, Intertwining methods in multi-dimensional scattering theory, I, preprint, Lund University, (1987).
19) M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56.
20) H. Pecher, L^p-Abschätzungen und klassiche Lösungen für nicht lineare Wellengleichungen, I, Math. Z., 150 (1976), 159-183.
21) H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.
22) M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Fourier Analysis, Selfadjointness, Academic Press, New York-San Francisco-London, 1975.
23) N. Shenk and D. Thoe, Outgoing solutions of (-Δ+q-k²)u=f in an exterior domain, J. Math. Anal. Appl., 31 (1970), 81-116.
24) R. S. Strichartz, A priori estimates for the wave equation and some applications, J. Funct. Anal., 5 (1970), 218-235.
25) R. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
26) M. Taylor, Pseudo-differential Operators, Princeton Univ. Press, Princeton NJ., 1981.
27) G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Univ. Press, Cambridge, 1922.
28) K. Yajima, On smoothing property of Schrödinger propagators, Lecture Notes in Math., 1450, 1990, pp. 20-35.
29) K. Yajima, The W^k,p-continuity of wave operators for Schrödinger operators, Proc. Japan Acad., 69 (1993), 94 98.

Right : [1] R. A. Adams, Sobolev Spaces, Academic Press, New York-San Francisco-London, 1975.
[2] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Ser. IV, 2 (1975), 151-218.
[3] M. Beals and W. Strauss, L^p estimates for the wave equation with a potential, preprint, 1992.
[4] L. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer, Berlin-Heidelberg-New York, 1976.
[5] Ph. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations, 56 (1985), 310-344.
[6] J. Ginibre and M. Moulin, Hilbert space approach to quantum mechanical three body problem, Ann. Inst. H. Poincaré Sec. A, 21 (1974), 97-145.
[7] J. Ginibre and G. Velo, The global Cauchy problem for some non-linear Schrödinger equations, revisited, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 2 (1985), 309-327.
[8] A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.
[9] A. Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions, Results in L₂(R^m), m≥5, Duke Math. J., 47 (1980), 57-80.
[10] A. Jensen and S. Nakamura, Mapping properties of functions of Schrödinger operators between L^p-spaces and Besov spaces, preprint, 1992.
[11] J. -L. Journe, A. Soffer and C. D. Sogge, Decay estimated for Schrödinger operators, Comm. Pure. Appl. Math., 44 (1991), 573-604.
[12] T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math., 12 (1959), 403-422.
[13] T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279.
[14] T. Kato and S. T. Kuroda, Theory of simple scattering and eigenfunction expansions, Functional analysis and related fields, Springer-Verlag, Berlin-Heidelberg-New York, 1970, pp. 99-131.
[15] T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Reviews in Math. Phys., 1 (1989), 481-496.
[16] S. T. Kuroda, An introduction to scattering theory, Lecture Notes Series, 51, Aarhus University, 1978, Aarhus, Denmark.
[17] S. T. Kuroda, Scattering theory for differential operators, I and II, J. Math. Soc. Japan, 25 (1972), 75-104 and 222-234.
[18] A. Melin, Intertwining methods in multi-dimensional scattering theory, I, preprint, Lund University, (1987).
[19] M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56.
[20] H. Pecher, L^p-Abschätzungen und klassiche Lösungen für nicht lineare Wellengleichungen, I, Math. Z., 150 (1976), 159-183.
[21] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.
[22] M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Fourier Analysis, Selfadjointness, Academic Press, New York-San Francisco-London, 1975.
[24] R. S. Strichartz, A priori estimates for the wave equation and some applications, J. Funct. Anal., 5 (1970), 218-235.
[25] R. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
[26] M. Taylor, Pseudo-differential Operators, Princeton Univ. Press, Princeton NJ., 1981.
[27] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Univ. Press, Cambridge, 1922.
[28] K. Yajima, On smoothing property of Schrödinger propagators, Lecture Notes in Math., 1450, 1990, pp. 20-35.
[29] K. Yajima, The W^k,p-continuity of wave operators for Schrödinger operators, Proc. Japan Acad., 69 (1993), 94-98.

訂正日: 2006/10/20 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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