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Inverse problem of quantum scattering theory. II. (English) Zbl 0373.35014
Translation from Itogi Nauki Tehn., Ser. Sovrem. Probl. Mat. 3, 93–180 (1974; Zbl 0299.35027).

81U40 Inverse scattering problems in quantum theory
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
34Lxx Ordinary differential operators
35P25 Scattering theory for PDEs
47A40 Scattering theory of linear operators
Full Text: DOI
[1] Z. S. Agranovich and V. A. Marchenko, Inverse Problem of Scattering Theory [in Russian], KGU, Kharkov (1960). · Zbl 0098.06004
[2] V. I. Arnol’d, Lectures on Classical Mechanics [in Russian], MGU, Moscow (1968).
[3] Yu. M. Berezanskii, ?Uniqueness theorem in the inverse problem of spectral analysis for the Schröedinger equation,? Tr. Moscow Matem. O-va,7, 3?62 (1958).
[4] V. S. Buslaev, ?Trace formulas for Schröedinger operator in three-dimensional space,? Dokl. Akad. Nauk SSSR,143, No. 5, 1067?1070 (1962).
[5] V. S. Buslaev and L. D. Faddeev, ?Trace formulas for the Sturm-Liouville differential singular operator,? Dokl. Akad. Nauk SSSR,132, No. 1, 13?16 (1960). · Zbl 0129.06501
[6] V. S. Buslaev and V. L. Fomin, Inverse Scattering Problem for One-Dimensional Schröedinger Equation on the Entire Axis, Vestn. Leningrad Un-ta, No. 1, (1962), pp. 56?64.
[7] M. G. Gasymov, ?Inverse problem of scattering theory for a system of Dirac equations of order 2n,? Tr. Moscow Matem. O-va,?19, 41?112 (1968). · Zbl 0179.20602
[8] I. M. Gel’fand and B. M. Levitan, ?Determination of a differential equation in terms of its spectral function,? Izv. Akad. Nauk SSSR. Ser. Matem.,15, No. 2, 309?360 (1951).
[9] I. M. Gel’fand and B. M. Levitan, ?Simple identity for eigenvalues of a second-order differential operator,? Dokl. Akad. Nauk SSSR,88, No. 4, 593?596 (1953).
[10] V. E. Zakharov and L. D. Faddeev, ?Korteweg-de Vries equation ? a completely integrable Hamiltonian system,? Funktional’. Analiz. i Ego Prilozhen.,5, No. 4, 18?27 (1971).
[11] V. E. Zakharov and A. B. Shabat, ?Rigorous theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,? Zh. Éksperim. i Teor. Fiz.,61, No. 1, 118?134 (1971).
[12] M. G. Krein, ?Determining particle potential by its S-function,? Dokl. Akad. Nauk SSSR,105, No. 3, 433?436 (1955).
[13] M. G. Krein and F. É. Melik-Adamyan, ?Theory of S-matrices of canonical differential equations with summable potential,? Dokl. Akad. Nauk ArmSSR,46, No. 4, 150?155 (1968).
[14] P. P. Kulish, ?Inverse scattering problem for Schröedinger equation on the axis,? Matem. Zametki,4, No. 6, 677?684 (1968). · Zbl 0176.04501
[15] B. Ya. Levin, ?Fourier-type and Laplace-type transformations by means of solutions of a second-order differential equation,? Dokl. Akad. Nauk,106, No. 2, 187?190 (1956).
[16] V. A. Marchenko, ?Reconstruction of potential energy in terms of stray wave phases,? Dokl. Akad. Nauk SSSR,104, No. 5, 695?698 (1955).
[17] V. A. Marehenko, Spectral Theory of Sturm-Liouville Operators [in Russian], Nauka Dumka, Kiev (1972).
[18] A. Ya. Povzner, ?Decomposition of arbitrary functions in eigenfunctions of the operator ?u+cu,? Matem. Sb.,32, No. 1, 109?156 (1953).
[19] A. Ya. Povzner, ?Decomposition in functions that are solutions of the scattering problem,? Dokl. Akad. Nauk SSSR,104, No. 3, 360?363 (1955).
[20] L. A. Takhtadzhyan, Method of the Inverse Problem for Solving the One-Dimensional Nonlinear Schröedinger Equation (Thesis), Matem. Mekhan. Dept., LGU (1972).
[21] L. D. Faddeev, Uniqueness of the Solution of Inverse Scattering Problem, Vestn. LGU, No. 7 (1956), pp. 126?130.
[22] L. D. Faddeev, Decomposition of Arbitrary Functions in Eigenfunctions of the Schröedinger Operator, Vestn. LGU, No. 7 (1957), pp. 164?172.
[23] L. D. Faddeev, ?Relation of S-matrix and the potential for the one-dimensional Schröedinger operator,? Dokl. Akad. Nauk SSSR,121, No. 1, 63?66 (1958). · Zbl 0085.43302
[24] L. D. Faddeev, ?Dispersion relations in nonrelativistic scattering theory,? Zh. Éksperim. i Teor. Fiz.,35, No. 2, 433?439 (1958).
[25] L. D. Faddeev, ?Inverse problem of quantum scattering theory,? Usp. Matem. Nauk,14, No. 4, 57?119 (1959). · Zbl 0091.21902
[26] L. D. Faddeev, ?Properties of the S-matrix of the one-dimensional Schröedinger equation,? Tr. Matem. In-ta Akad. Nauk SSSR,73, 314?336 (1964).
[27] L. D. Faddeev, ?Growing solutions of the Schröedinger equation,? Dokl. Akad. Nauk SSSR,165, No. 3, 514?517 (1965).
[28] L. D. Faddeev, ?Factorization of an S-matrix of a multidimensional Schröedinger operator,? Dokl. Akad. Nauk SSSR,167, No. 1, 69?72 (1966).
[29] L. D. Faddeev, ?Three-dimensional inverse problem of quantum scattering theory,? Sb. Tr. All-Union Symposium on Inverse Problems for Differential Equations [in Russian], Novosibirsk (1972).
[30] I. S. Frolov, ?Inverse scattering problem for Dirac system on the entire axis,? Dokl. Akad. Nauk SSSR,207, No. 1, 44?47 (1972). · Zbl 0284.34027
[31] F. Calogero and A. Degasperis, ?Values of the potential and its derivatives at the origin in terms of the s-wave phase shift and bound-state parameters,? J. Math. Phys.,9, No. 1, 90?116 (1968). · doi:10.1063/1.1664482
[32] O. D. Corbella, ?Inverse scattering problem for Dirac particles,? J. Math. Phys.,11, No. 5, 1695?1713 (1970). · doi:10.1063/1.1665315
[33] A. Degasperis, ?On the inverse problem for the Klein-Gordon s-wave equation,? J. Math. Phys.,11, No. 2, 551?567 (1970). · doi:10.1063/1.1665170
[34] T. Ikebe, ?Eigenfunction expansions associated with Schröedinger operators and their applications to scattering theory,? Arch. Ration. Mech. and Anal.,5, No. 1, 1?34 (1960). · Zbl 0145.36902 · doi:10.1007/BF00252896
[35] T. Kato, ?Growth properties of solutions of the reduced wave equation with a variable coefficient,? Communs. Pure and Appl. Math.,12, No. 3, 402?425 (1959). · Zbl 0091.09502 · doi:10.1002/cpa.3160120302
[36] T. Kato, Perturbation Theory for Linear Operators, Vol. 19, Springer, Berlin (1966). · Zbl 0148.12601
[37] I. Kay, ?The inverse scattering problem when the reflection coefficient is a rational function,? Communs. Pure and Appl. Math.,13, No. 3, 371?393 (1960). · Zbl 0093.20902 · doi:10.1002/cpa.3160130304
[38] I. Kay and H. E. Moses, ?The determination of the scattering potential from the spectral measure function. I,? Nuovo Cimento,2, No. 5, 917?961 (1955). · Zbl 0074.22501 · doi:10.1007/BF02855840
[39] I. Kay and H. E. Moses, ?The determination of the scattering potential from the spectral measure function. II,? Nuovo Cimento,3, No. 1, 66?84 (1956). · doi:10.1007/BF02746196
[40] I. Kay and H. E. Moses, ?The determination of the scattering potential from the spectral measure function. III,? Nuovo Cimento,3, No. 2, 276?304 (1956). · doi:10.1007/BF02745417
[41] N. N. Khuri, ?Analicity of the Schröedinger scattering amplitude and nonrelativistic dispersion relations,? Phys. Rev.,107, No. 4, 1148?1156 (1957). · Zbl 0079.42503 · doi:10.1103/PhysRev.107.1148
[42] M. D. Kruskal, C. S. Gardner, J. M. Greene, and R. M. Miura, ?Method for solving the Korteweg-de Vries equation,? Phys. Rev. Lett.,19, No. 19, 1095?1097 (1967). · Zbl 1061.35520 · doi:10.1103/PhysRevLett.19.1095
[43] M. D. Kruskal, R. M. Miura, C. S. Gardner, and N. J. Zabusky, ?Korteweg-de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws,? J. Math. Phys.,11, No. 3, 952?960 (1970). · Zbl 0283.35022 · doi:10.1063/1.1665232
[44] P. D. Lax, ?Integrals of nonlinear equations of evolution and solitary waves,? Communs. Pure and Appl. Math.,21, No. 5, 467?490 (1968). · Zbl 0162.41103 · doi:10.1002/cpa.3160210503
[45] P. D. Lax and R. Phillips, Scattering Theory, Vol. 12, Academic Press, New York-London (1967).
[46] N. Levinson, ?On the uniqueness of the potential in a Schröedinger equation for a given asymptotic phase,? Kgl. Danske Videnskab. Selskab. mat. Fys. medd.,25, No. 9 (1949).
[47] J. J. Loeffel, ?On an inverse problem in potential scattering theory,? Ann. Inst. H. Poincaré,8A, No. 4, 339?447 (1968).
[48] D. Wong, ?Dispersion relation for nonrelativistic particles,? Phys. Rev.,107, No. 1, 302?306 (1957). · Zbl 0087.43301 · doi:10.1103/PhysRev.107.302
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