Long-range potential scattering by Enss’s method in two Hilbert spaces

White, Denis A. W.

doi:10.1090/S0002-9947-1986-0831186-X

Long-range potential scattering by Enss’s method in two Hilbert spaces
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by Denis A. W. White PDF

Trans. Amer. Math. Soc. 295 (1986), 1-33 Request permission

Abstract:

Existence and completeness of wave operators is established by a straightforward transposition of the original short range result of Enss into an appropriate two-Hilbert space setting. Applied to long range quantum mechanical potential scattering, this result in conjunction with recent work of Isozaki and Kitada reduces the problem of proving existence and completeness of wave operators to that of approximating solutions of certain partial differential equations on cones in phase space. As an application existence and completeness of wave operators is established for Schrödinger operators with a long range multiplicative and possibly rapidly oscillating potential.

References

Some new results in spectral and scattering theory of differential operators in

Alberto-P. Calderón and Rémi Vaillancourt, A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 1185–1187. MR 298480, DOI 10.1073/pnas.69.5.1185
Monique Combescure, Spectral and scattering theory for a class of strongly oscillating potentials, Comm. Math. Phys. 73 (1980), no. 1, 43–62. MR 573612
Allen Devinatz and Peter Rejto, A limiting absorption principle for Schrödinger operators with oscillating potentials. I. $-\Delta +c\,\textrm {sin}(b\mid x\mid ^{\alpha })/\mid x\mid ^{\beta }$ for certain $c,\alpha ,\beta$, J. Differential Equations 49 (1983), no. 1, 29–84. MR 704264, DOI 10.1016/0022-0396(83)90019-0
Allen Devinatz and Peter Rejto, A limiting absorption principle for Schrödinger operators with oscillating potentials. I. $-\Delta +c\,\textrm {sin}(b\mid x\mid ^{\alpha })/\mid x\mid ^{\beta }$ for certain $c,\alpha ,\beta$, J. Differential Equations 49 (1983), no. 1, 29–84. MR 704264, DOI 10.1016/0022-0396(83)90019-0
John D. Dollard, Asymptotic convergence and the Coulomb interaction, J. Mathematical Phys. 5 (1964), 729–738. MR 163620, DOI 10.1063/1.1704171
John D. Dollard, Quantum-mechanical scattering theory for short-range and Coulomb interactions, Rocky Mountain J. Math. 1 (1971), no. 1, 5–88. MR 270673, DOI 10.1216/RMJ-1971-1-1-5
John D. Dollard and Charles N. Friedman, Existence of the Møller wave operators for $V(r)=(\lambda \textrm {sin}(\mu r^{\alpha })/r^{\beta })$, Ann. Physics 111 (1978), no. 1, 251–266. MR 489510, DOI 10.1016/0003-4916(78)90230-0
Volker Enss, Asymptotic completeness for quantum mechanical potential scattering. I. Short range potentials, Comm. Math. Phys. 61 (1978), no. 3, 285–291. MR 523013
Volker Enss, Asymptotic completeness for quantum-mechanical potential scattering. II. Singular and long-range potentials, Ann. Physics 119 (1979), no. 1, 117–132. MR 535624, DOI 10.1016/0003-4916(79)90252-5

Geometric methods in spectral and scattering theory of Schrödinger operators

Volker Enss, Asymptotic observables on scattering states, Comm. Math. Phys. 89 (1983), no. 2, 245–268. MR 709466
Volker Enss, Propagation properties of quantum scattering states, J. Funct. Anal. 52 (1983), no. 2, 219–251. MR 707205, DOI 10.1016/0022-1236(83)90083-6

La méthode ’dépendant du temps’ dans le problème de la complétude asymptotique

Lars Hörmander, The existence of wave operators in scattering theory, Math. Z. 146 (1976), no. 1, 69–91. MR 393884, DOI 10.1007/BF01213717
Teruo Ikebe and Hiroshi Isozaki, Completeness of modified wave operators for long-range potentials, Publ. Res. Inst. Math. Sci. 15 (1979), no. 3, 679–718. MR 566076, DOI 10.2977/prims/1195187871
Teruo Ikebe and Hiroshi Isozaki, A stationary approach to the existence and completeness of long-range wave operators, Integral Equations Operator Theory 5 (1982), no. 1, 18–49. MR 646878, DOI 10.1007/BF01694028
Hiroshi Isozaki, On the generalized Fourier transforms associated with Schrödinger operators with long-range perturbations, J. Reine Angew. Math. 337 (1982), 18–67. MR 676041, DOI 10.1515/crll.1982.337.18
Hiroshi Isozaki, Eikonal equations and spectral representations for long-range Schrödinger Hamiltonians, J. Math. Kyoto Univ. 20 (1980), no. 2, 243–261. MR 582166, DOI 10.1215/kjm/1250522277
Hiroshi Isozaki and Hitoshi Kitada, Modified wave operators with time-independent modifiers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 1, 77–104. MR 783182

Scattering matrices for two-body Schrödinger operators

Hiroshi Isozaki and Hitoshi Kitada, Asymptotic behavior of the scattering amplitude at high energies, Differential equations (Birmingham, Ala., 1983) North-Holland Math. Stud., vol. 92, North-Holland, Amsterdam, 1984, pp. 327–334. MR 799366, DOI 10.1016/S0304-0208(08)73711-3
Takashi Kako, Existence and equivalence of two types of long-range modified wave operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1978), no. 1, 133–147. MR 494599
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Hitoshi Kitada, Scattering theory for Schrödinger operators with long-range potentials. I. Abstract theory, J. Math. Soc. Japan 29 (1977), no. 4, 665–691. MR 634802, DOI 10.2969/jmsj/02940665
Hitoshi Kitada, Scattering theory for Schrödinger operators with long-range potentials. I. Abstract theory, J. Math. Soc. Japan 29 (1977), no. 4, 665–691. MR 634802, DOI 10.2969/jmsj/02940665
Hitoshi Kitada, Scattering theory for Schrödinger equations with time-dependent potentials of long-range type, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 353–369. MR 672067
Hitoshi Kitada, A calculus of Fourier integral operators and the global fundamental solution for a Schrödinger equation, Osaka J. Math. 19 (1982), no. 4, 863–900. MR 687775
Hitoshi Kitada, Time-decay of the high energy part of the solution for a Schrödinger equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 1, 109–146. MR 743522
Hitoshi Kitada and Kenji Yajima, A scattering theory for time-dependent long-range potentials, Duke Math. J. 49 (1982), no. 2, 341–376. MR 659945
Hitoshi Kitada and Kenji Yajima, Remarks on our paper: “A scattering theory for time-dependent long-range potentials” [Duke Math. J. 49 (1982), no. 2, 341–376; MR0659945 (83i:35137)], Duke Math. J. 50 (1983), no. 4, 1005–1016. MR 726315, DOI 10.1215/S0012-7094-83-05042-1
Eric Mourre, Link between the geometrical and the spectral transformation approaches in scattering theory, Comm. Math. Phys. 68 (1979), no. 1, 91–94. MR 539739
Pl. Muthuramalingam and Kalyan B. Sinha, Asymptotic completeness in long range scattering. II, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 1, 57–87. MR 803195
D. B. Pearson, Scattering theory for a class of oscillating potentials, Helv. Phys. Acta 52 (1979), no. 4, 541–554 (1980). MR 566255
Peter A. Perry, Mellin transforms and scattering theory. I. Short range potentials, Duke Math. J. 47 (1980), no. 1, 187–193. MR 563375

Scattering theory by the Enss method

Peter A. Perry, Propagation of states in dilation analytic potentials and asymptotic completeness, Comm. Math. Phys. 81 (1981), no. 2, 243–259. MR 632760
E. Prugovečki, On time-dependent scattering theory for long-range interactions, Nuovo Cimento B (11) 4 (1971), 105–123 (English, with Italian and Russian summaries). MR 305759, DOI 10.1007/BF02737569
E. Prugovečki and J. Zorbas, Modified Lippmann-Schwinger equations for two-body scattering theory with long-range interactions, J. Mathematical Phys. 14 (1973), 1398–1409. MR 329507, DOI 10.1063/1.1666194
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Barry Simon, Phase space analysis of simple scattering systems: extensions of some work of Enss, Duke Math. J. 46 (1979), no. 1, 119–168. MR 523604
Yoshimi Sait\B{o}, Spectral representations for Schrödinger operators with long-range potentials, Lecture Notes in Mathematics, vol. 727, Springer, Berlin, 1979. MR 540891
Kalyan B. Sinha and Pl. Muthuramalingam, Asymptotic evolution of certain observables and completeness in Coulomb scattering. I, J. Funct. Anal. 55 (1984), no. 3, 323–343. MR 734802, DOI 10.1016/0022-1236(84)90003-X
Denis A. W. White, Schrödinger operators with rapidly oscillating central potentials, Trans. Amer. Math. Soc. 275 (1983), no. 2, 641–677. MR 682723, DOI 10.1090/S0002-9947-1983-0682723-7
E. B. Davies, On Enss’ approach to scattering theory, Duke Math. J. 47 (1980), no. 1, 171–185. MR 563374