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Asymptotic completeness in long range scattering. II. (English) Zbl 0584.47009
Asymptotic completeness is proved for \(-\Delta /2+W_ S(Q)+W_ L(Q)\) on \(L^ 2(R^ v)\), \(v\geq 3\). Here \(W_ S\) is a short range potential while \(W_ L\) is a \(C^ m\) long range potential for m large enough and \(W_ L\) behaves like \((1+| x|)^{-\alpha}\) at \(\infty\) for \(1/2<\alpha <1\). [For part I see J. Funct. Anal. 55, 323-343 (1984; Zbl 0531.47008)].

MSC:
47A40 Scattering theory of linear operators
35P25 Scattering theory for PDEs
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[1] W. O. AMREIN , J. M. JAUCH and K. B. SINHA , Scattering Theory in Quantum Mechanics , W. Benjamin, Reading, 1977 . MR 58 #14631 | Zbl 0376.47001 · Zbl 0376.47001
[2] P. ALSHOLM , Wave Operators for Long Range Scattering (J. Math. Anal. Appl., Vol. 59, 1977 , p. 550). MR 58 #1740 | Zbl 0359.47006 · Zbl 0359.47006
[3] L. HÖRMANDER , The Existence of Wave Operators in Scattering Theory , (Math. Zeit., Vol. 146, 1976 , pp. 69-91). Article | MR 52 #14691 | Zbl 0319.35059 · Zbl 0319.35059
[4] A. M. BERTHIER and P. COLLET , Wave Operators for Momentum Dependent long Range Potential (Ann. Inst. Henri-Poincaré, Section A, Vol. 27, 1977 , pp. 279-293). Numdam | MR 57 #2303 | Zbl 0346.47010 · Zbl 0346.47010
[5] W. O. AMREIN , Ph. A. MARTIN and B. MISRA , On the Asymptotic Condition of Scattering Theory (Helv. Phys. Acta, Vol. 43, 1970 , pp. 313-344). MR 46 #8586 | Zbl 0195.56101 · Zbl 0195.56101
[6] J. D. DOLLARD , Asymptotic Convergence and the Coulomb Interaction (J. Math. Phys., Vol. 5, 1964 , pp. 729-738). MR 29 #921
[7] R. LAVINE , Absolute Continuity of Positive Spectrum for Schrödinger Operators with Long Range Potentials (J. Func. Anal., Vol. 12, 1973 , pp. 30-54). MR 49 #7624 | Zbl 0246.47017 · Zbl 0246.47017
[8] T. IKEBE , Spectral Representations for Schrödinger Operators with Long Range Potentials. Perturbation by short range potentials (Publ. Res. Inst. Math. Sc., Vol. 11, 1976 , pp. 551-558). MR 55 #13092 | Zbl 0345.35032 · Zbl 0345.35032
[9] Y. SAITO , Spectral Representations for Schrödinger Operators with long Range potential (Lecture Notes in Math. No. 727, Berlin. Heildelberg, New York, Springer-Verlag, 1979 ). MR 81a:35083 | Zbl 0414.47012 · Zbl 0414.47012
[10] J. WEIDMANN , Über Spectraltheorie von SturmLiouville-Operatoren (Math. Zeit., Vol. 98, 1967 , pp. 268-302). Article | MR 35 #4769 | Zbl 0168.12301 · Zbl 0168.12301
[11] V. GEORGESCU , Méthodes stationnaires pour des potentiels à longue portée à symétrie sphérique (Thèse, Université de Genève, 1974 ).
[12] T. IKEBE and H. ISOZAKI , A Stationary Approach to the Existence and Completeness of Long Range wave Operators , preprint, Kyoto Univ., 1980 . · Zbl 0496.35069
[13] H. KITADA , Scattering Theory for Schrodinger Operators with Long Range Potentials II (J. Math. Soc. Japan, Vol. 30, 1978 , pp. 603-632). Article | MR 58 #30372b | Zbl 0388.35055 · Zbl 0388.35055
[14] S. AGMON , Some new Results in Spectral and Scattering Theory of Differential Operators on L\(^{2}\)(Rn) [Séminaire Goulaouic-Schwartz, 1978 - 1979 , Centre de Mathématiques, Palaiseau (Lecture notes)]. Numdam | Zbl 0406.35052 · Zbl 0406.35052
[15] L. E. THOMAS , On the Algebraic Theory of Scattering (J. Func. Anal., Vol. 15, 1974 , pp. 364-377). MR 54 #6783 | Zbl 0283.47006 · Zbl 0283.47006
[16] V. ENSS , Asymptotic Completeness for Quantum Mechanical Potential Scattering II. Singular and Long Range Potentials (Ann. Phys. New York, Vol. 119, 1979 , pp. 117-132). MR 80k:81144 | Zbl 0408.47009 · Zbl 0408.47009
[17] V. ENSS , Geometric Methods in Spectral and Scattering Theory of Schrödinger Operators , Section 7 in (Rigorous Atomic and Molecular Physics), G. VELO and A. WEIGHTMAN eds., New York, Plenum, 1981 .
[18] P. A. PERRY , Propagation of States in Dilation Analytic Potentials and Asymptotic Completeness (Comm. Math. Phys., Vol. 81, 1981 , pp. 243-259). Article | MR 84f:81097 | Zbl 0471.47007 · Zbl 0471.47007
[19] Pl. MUTHURAMALINGAM and K. B. SINHA , Asymptotic Evoluation of Certain Observables and Completeness in Coulomb Scattering I (J. Func. Analysis, Vol. 55, 1983 ). Zbl 0531.47008 · Zbl 0531.47008
[20] T. KATO , Perturbation Theory for Linear Operators , Springer, Berlin, 1966 . MR 34 #3324 | Zbl 0148.12601 · Zbl 0148.12601
[21] M. REED and B. SIMON , Methods of Modern Mathematical Physics, II . Fourier Analysis, Self Adjointness, Academic Press, New York, 1972 . Zbl 0308.47002 · Zbl 0308.47002
[22] M. REED and B. SIMON , Methods of Modern Mathematical Physics, III . Scattering Theory, Academic Press, New York, 1979 . MR 80m:81085 | Zbl 0405.47007 · Zbl 0405.47007
[23] M. REED and B. SIMON , Methods of Modern Mathematical Physics, IV . Analysis of Operators, Academic Press, New York, 1978 . MR 58 #12429c | Zbl 0401.47001 · Zbl 0401.47001
[24] P. A. PERRY , Mellin Transforms and Scattering Theory, I . Short Range Potentials (Duke Math. J., Vol. 47, 1980 , pp. 187-193). Article | MR 81c:35101 | Zbl 0445.47009 · Zbl 0445.47009
[25] E. B. DAVIES , Quantum Theory of Open Systems , Academic Press, New York, 1976 . MR 58 #8853 | Zbl 0388.46044 · Zbl 0388.46044
[26] E. B. DAVIES , On Enss’ Approach to Scattering Theory (Duke Math. J. Vol. 47, 1980 , pp. 171-185). Article | MR 81c:81046 | Zbl 0434.47014 · Zbl 0434.47014
[27] J. GINIBRE , La Méthode ”dépendant du temps” dans le problème de la complétude asymptotique , preprint, Université de Paris-Sud, LP THE 80/10, 1980 .
[28] E. MOURRE , Opérateurs conjugués et propriétés de propagation (Commun. Math. Phys., Vol. 91, 1983 , pp. 279-300). Article | MR 86h:47031 | Zbl 0543.47041 · Zbl 0543.47041
[29] A. JENSEN , E. MOURRE and P. PERRY , Multiple Commutator Estimates and Resolvent Smoothness in Quantum Scattering Theory , Preprint, 1983 , Caltech, CA 91125. · Zbl 0561.47007
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