A generalization of the radiation condition of Sommerfeld for $N$-body Schrödinger operators

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A generalization of the radiation condition of Sommerfeld for $N$-body Schrödinger operators

Hiroshi Isozaki

Source: Duke Math. J. Volume 74, Number 2 (1994), 557-584.

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Primary Subjects: 81U10

Secondary Subjects: 35P25, 47N50

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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077288208
Mathematical Reviews number (MathSciNet): MR1272984
Zentralblatt MATH identifier: 0811.35107
Digital Object Identifier: doi:10.1215/S0012-7094-94-07420-6

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References

[1] S. Agmon, Lower bounds for solutions of Schrödinger equations, J. Analyse Math. 23 (1970), 1–25.

Mathematical Reviews (MathSciNet): MR43:2367

Zentralblatt MATH: 0211.40703

Digital Object Identifier: doi:10.1007/BF02795485

[2] S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math. 30 (1976), 1–38.

Mathematical Reviews (MathSciNet): MR57:6776

Zentralblatt MATH: 0335.35013

Digital Object Identifier: doi:10.1007/BF02786703

[3] M. S. P. Eastham and H. Kalf, Schrödinger-type operators with continuous spectra, Research Notes in Mathematics, vol. 65, Pitman (Advanced Publishing Program), Boston, Mass., 1982.

Mathematical Reviews (MathSciNet): MR84i:35107

Zentralblatt MATH: 0491.35003

[4] D. M. Eidus, The principle of limit amplitude, Russian Math. Surveys 24 (1969), 97–167.

Zentralblatt MATH: 0197.08102

Mathematical Reviews (MathSciNet): MR601072

[5] R. Froese and I. Herbst, Exponential bounds and absence of positive eigenvalues for $N$-body Schrödinger operators, Comm. Math. Phys. 87 (1982/83), no. 3, 429–447.

Mathematical Reviews (MathSciNet): MR85g:35091

Zentralblatt MATH: 0509.35061

Digital Object Identifier: doi:10.1007/BF01206033

Project Euclid: euclid.cmp/1103922052

[6] R. Froese and I. Herbst, A new proof of the Mourre estimate, Duke Math. J. 49 (1982), no. 4, 1075–1085.

Mathematical Reviews (MathSciNet): MR85d:35092

Zentralblatt MATH: 0514.35025

Digital Object Identifier: doi:10.1215/S0012-7094-82-04947-X

Project Euclid: euclid.dmj/1077315539

[7] C. Gérard, Sharp propagation estimates for $N$-particle systems, Duke Math. J. 67 (1992), no. 3, 483–515.

Mathematical Reviews (MathSciNet): MR93j:35061

Zentralblatt MATH: 0760.35049

Digital Object Identifier: doi:10.1215/S0012-7094-92-06719-6

Project Euclid: euclid.dmj/1077294535

[8] C. Gérard, H. Isozaki, and E. Skibsted, Commutator algebra and resolvent estimates, to appear in Spectral and Scattering Theory and Related Topics, Adv. Stud. Pure Math. 23, Academic Press, Boston, 1994.

Mathematical Reviews (MathSciNet): MR1275395

[9] V. V. Grushin, On Sommerfeld-type conditions for a certain class of partial differential equations, Eleven Papers on Differential Equations, Functional Analysis and Measure Theory, Amer. Math. Soc. Transl. Ser. 2, vol. 51, Amer. Math. Soc., Providence, 1966, pp. 82–112.

[10] B. Helffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988) eds. H. Holden and A. Jensen, Lecture Notes in Phys., vol. 345, Springer, Berlin, 1989, pp. 118–197.

Mathematical Reviews (MathSciNet): MR91g:35078

Zentralblatt MATH: 0699.35189

[11] I. Herbst and E. Skibsted, Time-dependent approach to radiation conditions, Duke Math. J. 64 (1991), no. 1, 119–147.

Mathematical Reviews (MathSciNet): MR92k:47140

Zentralblatt MATH: 0756.35063

Digital Object Identifier: doi:10.1215/S0012-7094-91-06406-9

Project Euclid: euclid.dmj/1077295389

[12] L. Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (1979), no. 3, 360–444.

Mathematical Reviews (MathSciNet): MR80j:47060

Zentralblatt MATH: 0388.47032

[13] T. Ikebe and Y. Saito, Limiting absorption method and absolute continuity for the Schrödinger operator, J. Math. Kyoto Univ. 12 (1972), 513–542.

Mathematical Reviews (MathSciNet): MR47:628

Zentralblatt MATH: 0257.35022

Project Euclid: euclid.kjm/1250523478

[14] H. Isozaki, Asymptotic properties of generalized eigenfunctions for three body Schrödinger operators, Comm. Math. Phys. 153 (1993), no. 1, 1–21.

Mathematical Reviews (MathSciNet): MR94f:81172

Zentralblatt MATH: 0774.35064

Digital Object Identifier: doi:10.1007/BF02099038

Project Euclid: euclid.cmp/1104252594

[15] W. Jäger, Ein gewöhnlicher Differentialoperator zweiter Ordnung für#Funktionen mit Werten in einem Hilbertraum, Math. Z. 113 (1970), 68–98.

Mathematical Reviews (MathSciNet): MR57:6699

Zentralblatt MATH: 0187.09001

Digital Object Identifier: doi:10.1007/BF01110098

[16] T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12 (1959), 403–425.

Mathematical Reviews (MathSciNet): MR21:7349

Zentralblatt MATH: 0091.09502

Digital Object Identifier: doi:10.1002/cpa.3160120302

[17] K. Mochizuki, Growth properties of solutions of second order elliptic differential equations, J. Math. Kyoto Univ. 16 (1976), no. 2, 351–373.

Mathematical Reviews (MathSciNet): MR54:5603

Zentralblatt MATH: 0407.35031

Project Euclid: euclid.kjm/1250522919

[18] R. G. Newton, The asymptotic form of three-particle wavefunctions and the cross sections, Ann. Physics 74 (1972), 324–351.

Mathematical Reviews (MathSciNet): MR47:9978

Digital Object Identifier: doi:10.1016/0003-4916(72)90144-3

[19] P. Perry, I. M. Sigal, and B. Simon, Spectral analysis of $N$-body Schrödinger operators, Ann. of Math. (2) 114 (1981), no. 3, 519–567.

Mathematical Reviews (MathSciNet): MR83b:81129

Zentralblatt MATH: 0477.35069

Digital Object Identifier: doi:10.2307/1971301

JSTOR: links.jstor.org

[20] F. Rellich, Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u=0$ in unendlichen Gebieten, Jber. Deutsch. Math. Verein. 53 (1943), 57–65.

Mathematical Reviews (MathSciNet): MR8,204c

Zentralblatt MATH: 0028.16401

[21] S. N. Roze, The spectrum of a second order elliptic operator, Mat. Sb. (N.S.) 80 (122) (1969), 195–209.

Mathematical Reviews (MathSciNet): MR40:6090

Zentralblatt MATH: 0184.32903

[22] Y. Saito, Spectral representations for Schrödinger operators with long-range potentials, Lecture Notes in Mathematics, vol. 727, Springer, Berlin, 1979.

Mathematical Reviews (MathSciNet): MR81a:35083

Zentralblatt MATH: 0414.47012

[23] J. R. Schulenberger and C. H. Wilcox, A Rellich uniqueness theorem for steady-state wave propagation in inhomogeneous anisotropic media, Arch. Rational Mech. Anal. 41 (1971), 18–45.

Mathematical Reviews (MathSciNet): MR43:712

Zentralblatt MATH: 0218.35053

[24] A. Sommerfeld, Die Greensche Funktionen der Schwingungsgleichung, Jahresber. Deutsch. Math. Verein 21 (1912), 309–353.

[25] A. Sommerfeld, Partial Differential Equations in Physics, Academic Press Inc., New York, N. Y., 1949.

Mathematical Reviews (MathSciNet): MR10,608b

Zentralblatt MATH: 0034.35702

[26] J. Uchiyama, Lower bounds of growth order of solutions of Schrödinger equations with homogeneous potentials, Publ. Res. Inst. Math. Sci. 10 (1974/75), 425–444.

Mathematical Reviews (MathSciNet): MR51:8616

Zentralblatt MATH: 0321.35027

Digital Object Identifier: doi:10.2977/prims/1195192003

[27] B. R. Vainberg, Principles of radiation, limit absorption and limit amplitude, Russian Math. Surveys 21 (1966), 115–193.

Mathematical Reviews (MathSciNet): MR213701

[28] E. Vecoua, On metaharmonic functions, Trav. Inst. Math. Tbilissi [Trudy Tbiliss. Mat. Inst.] 12 (1943), 105–174.

Mathematical Reviews (MathSciNet): MR6,154i

[29] X. P. Wang, Micro-local resolvent estimates for $N$-body Schrödinger operators, to appear in J. Fac. Sci. Univ. Tokyo, Sect. IA.

[30] J. Weidmann, On the continuous spectrum of Schrödinger operators, Comm. Pure Appl. Math. 19 (1966), 107–110.

Mathematical Reviews (MathSciNet): MR33:413

Zentralblatt MATH: 0138.35804

[31] D. Yafaev, Radiation conditions and scattering theory for $N$-particle Hamiltonians, Comm. Math. Phys. 154 (1993), no. 3, 523–554.

Mathematical Reviews (MathSciNet): MR95b:81220

Zentralblatt MATH: 0781.35048

Digital Object Identifier: doi:10.1007/BF02102107

Project Euclid: euclid.cmp/1104253077

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