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Literature cited
Z. S. Agranovich and V. A. Marchenko, Inverse Problem of Scattering Theory [in Russian], KGU, Kharkov (1960).
V. I. Arnol'd, Lectures on Classical Mechanics [in Russian], MGU, Moscow (1968).
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).
V. S. Buslaev, “Trace formulas for Schröedinger operator in three-dimensional space,” Dokl. Akad. Nauk SSSR,143, No. 5, 1067–1070 (1962).
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).
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.
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).
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).
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).
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).
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).
M. G. Krein, “Determining particle potential by its S-function,” Dokl. Akad. Nauk SSSR,105, No. 3, 433–436 (1955).
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).
P. P. Kulish, “Inverse scattering problem for Schröedinger equation on the axis,” Matem. Zametki,4, No. 6, 677–684 (1968).
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).
V. A. Marchenko, “Reconstruction of potential energy in terms of stray wave phases,” Dokl. Akad. Nauk SSSR,104, No. 5, 695–698 (1955).
V. A. Marehenko, Spectral Theory of Sturm-Liouville Operators [in Russian], Nauka Dumka, Kiev (1972).
A. Ya. Povzner, “Decomposition of arbitrary functions in eigenfunctions of the operator Δu+cu,” Matem. Sb.,32, No. 1, 109–156 (1953).
A. Ya. Povzner, “Decomposition in functions that are solutions of the scattering problem,” Dokl. Akad. Nauk SSSR,104, No. 3, 360–363 (1955).
L. A. Takhtadzhyan, Method of the Inverse Problem for Solving the One-Dimensional Nonlinear Schröedinger Equation (Thesis), Matem. Mekhan. Dept., LGU (1972).
L. D. Faddeev, Uniqueness of the Solution of Inverse Scattering Problem, Vestn. LGU, No. 7 (1956), pp. 126–130.
L. D. Faddeev, Decomposition of Arbitrary Functions in Eigenfunctions of the Schröedinger Operator, Vestn. LGU, No. 7 (1957), pp. 164–172.
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).
L. D. Faddeev, “Dispersion relations in nonrelativistic scattering theory,” Zh. Éksperim. i Teor. Fiz.,35, No. 2, 433–439 (1958).
L. D. Faddeev, “Inverse problem of quantum scattering theory,” Usp. Matem. Nauk,14, No. 4, 57–119 (1959).
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).
L. D. Faddeev, “Growing solutions of the Schröedinger equation,” Dokl. Akad. Nauk SSSR,165, No. 3, 514–517 (1965).
L. D. Faddeev, “Factorization of an S-matrix of a multidimensional Schröedinger operator,” Dokl. Akad. Nauk SSSR,167, No. 1, 69–72 (1966).
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).
I. S. Frolov, “Inverse scattering problem for Dirac system on the entire axis,” Dokl. Akad. Nauk SSSR,207, No. 1, 44–47 (1972).
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).
O. D. Corbella, “Inverse scattering problem for Dirac particles,” J. Math. Phys.,11, No. 5, 1695–1713 (1970).
A. Degasperis, “On the inverse problem for the Klein-Gordon s-wave equation,” J. Math. Phys.,11, No. 2, 551–567 (1970).
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).
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).
T. Kato, Perturbation Theory for Linear Operators, Vol. 19, Springer, Berlin (1966).
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).
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).
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).
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).
N. N. Khuri, “Analicity of the Schröedinger scattering amplitude and nonrelativistic dispersion relations,” Phys. Rev.,107, No. 4, 1148–1156 (1957).
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).
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).
P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Communs. Pure and Appl. Math.,21, No. 5, 467–490 (1968).
P. D. Lax and R. Phillips, Scattering Theory, Vol. 12, Academic Press, New York-London (1967).
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).
J. J. Loeffel, “On an inverse problem in potential scattering theory,” Ann. Inst. H. Poincaré,8A, No. 4, 339–447 (1968).
D. Wong, “Dispersion relation for nonrelativistic particles,” Phys. Rev.,107, No. 1, 302–306 (1957).
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Translated from Itogi Nauki i Tekhniki. Sovremennye Problemy Matematiki, Vol. 3, pp. 93–180, 1974.
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Faddeev, L.D. Inverse problem of quantum scattering theory. II.. J Math Sci 5, 334–396 (1976). https://doi.org/10.1007/BF01083780
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DOI: https://doi.org/10.1007/BF01083780