Article PDF
Avoid common mistakes on your manuscript.
Literature Cited
V. A. Marchenko, The Sturm-Liouville Operators and Their Applications [in Russian], Naukova Dumka, Kiev (1977).
K. Chadan and P. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York-Berlin (1977).
B. M. Levitan, The Sturm-Liouville Inverse Problems [in Russian], Nauka, Moscow (1984).
Yu. M. Berezanskii, "On the uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation," Tr. Mosk. Mat. Obshch.,7, 3–62 (1958).
H. E. Moses, "Calculation of the scattering potential from reflection coefficients," Phys. Rev.,102, 559–597 (1956).
L. D. Faddeev, "The uniqueness of solution of the inverse scattering problem," Vestn. Leningr. Gos. Univ., No. 7, 126–230 (1956).
M. M. Lavrent'ev, "On an inverse problem for the wave equation," Dokl. Akad. Nauk SSSR,157, No. 3, 520–521 (1964).
L. D. Faddeev, "Growing solutions of the Schrödinger equation," Dokl. Akad. Nauk SSSR,165, No. 3, 514–517 (1965).
L. D. Faddeev, "The inverse problem in the quantum theory of scattering. II," in: Current Problems in Mathematics [in Russian], Vol. 3, Akad. Nauk SSSR, Vsesoyuznyi Inst. Nauchnoi i Tekhnicheskoi Informatsii, Moscow (1974), pp. 93–180.
R. G. Newton, "The Gelfand-Levitan method in the inverse scattering problems," in: Scattering Theory in Mathematical Physics, J. A. Lavita and J. P. Marchande (eds.), Reidel Publ. Co., Dordrecht (1974), pp. 193–235.
B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, "The Schrödinger equation in a periodic field and Riemann surfaces," Dokl. Akad. Nauk SSSR,229, No. 1, 15–18 (1976).
V. G. Romanov, Inverse Problems of Mathematical Physics [in Russian], Nuaka, Moscow (1984).
R. G. Newton, "The Marchenko and Gel'fand—Levitan methods in the inverse scattering problem in one and three dimensions," in: Conference on Inverse Scattering: Theory and Application, J. B. Bednaretal (ed.), SIAM (Society for Industrial and Applied Mathematics), Philadelphia (1983), pp. 1–74.
A. S. Fokas and M. J. Ablowitz, "On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane," J. Math. Phys.,25, 2494–2505 (1984).
A. P. Veselov and S. P. Novikov, "Finite-zone two-dimensional Schrödinger operators. Identification of potential operators. Real theory," Dokl. Akad. Nauk SSSR,279, No. 4, 784–788 (1984).
A. I. Nachman and M. J. Ablowitz, "A multidimensional inverse scattering method," Stud. Appl. Math.,71, 243–250 (1984).
R. Beals and R. R. Coifman, "Multidimensional scattering and nonlinear partial differential equations," Proc. of Symposia in Pure Mathematics,43, 45–70 (1985).
P. G. Grinevich and R. G. Novikov, "Analogues of multisoliton potentials for the two-dimensional Schrödinger operator, and a nonlocal Riemann problem," Dokl. Akad. Nauk SSSR,286, No. 1, 19–22 (1986).
R. G. Novikov, "Construction of a two-dimensional Schrödinger operator with a given scattering amplitude at a fixed energy," Teor. Mat. Fiz.,66, No. 2, 234–240 (1986).
P. G. Grinevich and S. V. Manakov, "The inverse problem in scattering theory for the two-dimensional Schrödinger operator,\(\bar \partial \)-method, and nonlinear equations," Funkts. Anal. Prilozhen.,20, No. 2, 14–24 (1986).
R. G. Novikov, "Reconstruction of a two-dimensional Schrödinger operator from the scattering amplitude at a fixed energy," Funkts. Anal. Prilozhen.,20, No. 3, 90–91 (1986).
A. S. Fokas, "An inverse problem for multidimensional first-order systems," J. Math. Phys.,27, 1737–1746 (1986).
R. G. Novikov and G. M. Khenkin, "The\(\bar \partial \)-equation in the multidimensional inverse scattering problem," Preprint 27M, Institute of Physics, Krasnoyarsk (1986). Usp. Mat. Nauk,42, No. 3, 93–152 (1987).
G. M. Henkin and R. G. Novikov, "A multidimensional inverse problem in quantum and acoustic scattering," Inverse Problems,4, 103–121 (1988).
P. G. Grinevich and S. P. Novikov, "Two-dimensional ‘inverse scattering problem’ for negative energies, and generalized-analytic functions. I. Energies below the ground state," Funkts. Anal. Prilozhen.,22, No. 1, 23–33 (1978).
I. M. Gel'fand, "Some problems of functional analysis and algebra," in: International Mathematical Congress in Amsterdam [in Russian], Nauka, Moscow (1961), pp. 49–74.
M. I. Belishev, "On an approach to multidimensional inverse problems for the wave equation," Dokl. Akad. Nauk SSSR,297, No. 3, 523–526 (1987).
J. Sylvester and G. Uhlmann, "A uniqueness theorem for an inverse boundary value problem in electrical prospection," Commun. Pure Appl. Math.,39, 91–112 (1986).
J. Sylvester and G. Uhlmann, "A global uniqueness theorem," Ann. Math.,125, 153–169 (1987).
A. P. Calderon, "On an inverse boundary value problem," in: Seminar on Numerical Analysis and Its Applications to Continuum Physics, Soc. Brasileira de Matematica, Rio de Janeiro (1980).
R. Kohn and M. Vogelius, "Determining conductivity by boundary measurements. II. Interior results," Commun. Pure Appl. Math.,38, 643–667 (1985).
Additional information
M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 4, pp. 11–22, October–December, 1988.
Rights and permissions
About this article
Cite this article
Novikov, R.G. Multidimensional inverse spectral problem for the equation —Δψ + (v(x) — Eu(x))ψ = 0. Funct Anal Its Appl 22, 263–272 (1988). https://doi.org/10.1007/BF01077418
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01077418