Scattering theory for Schrödinger operators with long-range potentials, I, abstract theory

Hitoshi KITADA

doi:10.2969/jmsj/02940665

Hitoshi KITADA

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JOURNAL FREE ACCESS

1977 Volume 29 Issue 4 Pages 665-691

DOI https://doi.org/10.2969/jmsj/02940665

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Published: 1977 Received: March 08, 1976 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, (4) 2 (1975), 151-218.
2) P. Alsholm, Wave operators for long-range scattering, (preprint, 1976).
3) P. Alsholm and T. Kato, Scattering with long-range potentials, Proc. Symp. Pure Math., 23 (1973), 393-399.
4) W. O. Amrein, Ph. A. Martin and B. Misra, On the asymptotic condition of scattering theory, Helv. Phys. Acta, 43 (1970), 313-344.
5) V.S. Buslaev and V. B. Matveev, Wave operators for the Schrödinger equation with a slowly decreasing potential, Theoret, and Math. Phys., 2 (1970), 266-274 (English translation from Russian).
6) C. Chandler and A. G. Gibson, Invariance principle for scattering with long-range (and other) potentials, Indiana Univ. Math. J., 25 (1976), 443-460.
7) J. D. Dollard, Asymptotic convergence and the Coulomb interaction, J. Math. Phys., 5 (1964), 729-738.
8) L. Hörmander, The existence of wave operators in scattering theory, Math. Z., 146 (1976), 69-91.
9) T. Ikebe, Spectral representation for Schrödinger operators with long-range potentials, J. Functional Analysis, 20 (1975), 158-177.
10) T. Ikebe and Y. Saito, Limiting absorption method and absolute continuity for the Schrödinger operator, J. Math. Kyoto Univ., 12 (1972), 513-542.
11) H. Isozaki, On the long-range stationary wave operator, (to appear).
12) T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, Heiderberg, and New York, 1976.
13) T. Kato and S. T. Kuroda, Theory of simple scattering and eigenfunction expansions, Functional Analysis and Related Fields, Springer-Verlag, Berlin, Heiderberg, and New York, 1970, 99-131.
14) T. Kato and S. T. Kuroda, The abstract theory of scattering, Rocky Mount. J. Math., 1 (1971), 127-171.
15) H. Kitada, A stationary approach to long-range scattering, Osaka J. Math., 13 (1976), 311-333.
16) H. Kitada, On the completeness of modified wave operators, Proc. Japan Acad., 52 (1976), 409-412.
17) S. T. Kuroda, Scattering theory for differential operators, I, operator theory, J. Math. Soc. Japan, 25 (1973), 75-104.
18) S. T. Kuroda, Scattering theory for differential operators, II, self-adjoint elliptic operators, J. Math. Soc. Japan, 25 (1973), 222-234.
19) R. Lavine, Absolute continuity of positive spectrum for Schrödinger operators with long-range potentials, J. Functional Analysis, 12 (1973), 30-54.
20) R. Lavine, Commutators and scattering theory, II, Indiana Univ. Math. J., 21 (1972), 643-656.
21) V. B. Matveev, The invariance principle for generalized wave operators, Topics in Mathematics, Consultants Bureau, New York, London, 1972, 77-85 (English translation from Russian).
22) G. Pinchuk, Abstract time-independent wave operator theory for long-range potentials, (preprint, 1975).
23) Y. Saito, The principle of limiting absorption for second-order differential equations with operator-valued coefficients, Publ. RIMS, Kyoto Univ., 7 (1971/72), 581-619.
24) Y. Saito, Spectral and scattering theory for second-order differential operators with operator-valued coefficients, Osaka J. Math., 9 (1972), 463-498.
25) Y. Saito, Spectral theory for second-order differential operators with long-range operator-valued coefficients, I, limiting absorption principle, Japan. J. Math., 1 (1975), 311-349.
26) Y. Saito, Spectral theory for second-order differential operators with long-range operator-valued coefficients, II, eigenfunction expansions and the Schrödinger operators with long-range potentials, Japan. J. Math., 1 (1975), 351-382.

Right : [1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa, (4) 2 (1975), 151-218.
[2] P. Alsholm, Wave operators for long-range scattering, (preprint, 1976).
[3] P. Alsholm and T. Kato, Scattering with long-range potentials, Proc. Symp. Pure Math., 23 (1973), 393-399.
[4] W. O. Amrein, Ph. A. Martin and B. Misra, On the asymptotic condition of scattering theory, Helv. Phys. Acta, 43 (1970), 313-344.
[5] V. S. Buslaev and V. B. Matveev, Wave operators for the Schrödinger equation with a slowly decreasing potential, Theoret, and Math. Phys., 2 (1970), 266-274 (English translation from Russian).
[6] C. Chandler and A. G. Gibson, Invariance principle for scattering with long-range (and other) potentials, Indiana Univ. Math. J., 25 (1976), 443-460.
[7] J. D. Dollard, Asymptotic convergence and the Coulomb interaction, J. Math. Phys., 5 (1964), 729-738.
[8] L. Hörmander, The existence of wave operators in scattering theory, Math. Z., 146 (1976), 69-91.
[9] T. Ikebe, Spectral representation for Schrödinger operators with long-range potentials, J. Functional Analysis, 20 (1975), 158-177.
[10] T. Ikebe and Y. Saito, Limiting absorption method and absolute continuity for the Schrödinger operator, J. Math. Kyoto Univ., 12 (1972), 513-542.
[11] H. Isozaki, On the long-range stationary wave operator, (to appear).
[12] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, Heiderberg, and New York, 1976.
[13] T. Kato and S. T. Kuroda, Theory of simple scattering and eigenfunction expansions, Functional Analysis and Related Fields, Springer-Verlag, Berlin, Heiderberg, and New York, 1970, 99-131.
[14] T. Kato and S. T. Kuroda, The abstract theory of scattering, Rocky Mount. J. Math., 1 (1971), 127-171.
[15] H. Kitada, A stationary approach to long-range scattering, Osaka J. Math., 13 (1976), 311-333.
[16] H. Kitada, On the completeness of modified wave operators, Proc. Japan Acad., 52 (1976), 409-412.
[17] S. T. Kuroda, Scattering theory for differential operators, I, operator theory, J. Math. Soc. Japan, 25 (1973), 75-104.
[18] S. T. Kuroda, Scattering theory for differential operators, II, self-adjoint elliptic operators, J. Math. Soc. Japan, 25 (1973), 222-234.
[19] R. Lavine, Absolute continuity of positive spectrum for Schrödinger operators with long-range potentials, J. Functional Analysis, 12 (1973), 30-54.
[20] R. Lavine, Commutators and scattering theory, II, Indiana Univ. Math. J., 21 (1972), 643-656.
[21] V. B. Matveev, The invariance principle for generalized wave operators, Topics in Mathematics, Consultants Bureau, New York, London, 1972, 77-85 (English translation from Russian).
[22] G. Pinchuk, Abstract time-independent wave operator theory for long-range potentials, (preprint, 1975).
[23] Y. Saito, The principle of limiting absorption for second-order differential equations with operator-valued coefficients, Publ. RIMS, Kyoto Univ., 7 (1971/72), 581-619.
[24] Y. Saito, Spectral and scattering theory for second-order differential operators with operator-valued coefficients, Osaka J. Math., 9 (1972), 463-498.
[25] Y. Saito, Spectral theory for second-order differential operators with long-range operator-valued coefficients, I, limiting absorption principle, Japan. J. Math., 1 (1975), 311-349.
[26] Y. Saito, Spectral theory for second-order differential operators with long-range operator-valued coefficients, II, eigenfunction expansions and the Schrödinger operators with long-range potentials, Japan. J. Math., 1 (1975), 351-382.
[27] Y. Saito, On the asymptotic behavior of the solutions of the Schrödinger equation (-Δ+Q(y)-k²)V=F, Osaka J. Math., 14 (1977), 11-35.
[28] Y. Saito, Eigenfunction expansions for the Schrödinger operators with long-range potentials Q(y)=0(|y|^-ε), ε>0, Osaka J. Math., 14 (1977), 37-53.

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