Article PDF
Avoid common mistakes on your manuscript.
References
Berezanski, Yu. M.: Expansion in terms of the eigenfunctions of self-adjoint operators [in Russian]. Mat. Sbornik 48 (85), 75–126 (1957).
Browder, F. E.: The eigenfunction expansion theorem for the general selfadjoint elliptic partial differential operator, I and II. Proc. Nat. Acad. Sci. U.S.A. 40, No 6, 454–459, 459–463 (1954).
Browder, F. E.: Eigenfunction expansions for formally self-adjoint partial differential operators, I and II. Proc. Nat. Acad. Sci. U.S.A. 42, No. 10, 769–771; No. 11, 870–872 (1956).
Browder, F. E.: Eigenfunction expansions for non-symmetric partial differential operators, I and II. Amer. J. Math. 80, 365–381 (1958); 81, 1–22 (1959).
Burnat, M.: On the solution of a problem for the Schroedinger equation in the infinite 3-dimensional space [in Russian]. Dokl. Akad. Nauk SSSR. 112, 224–227 (1957).
Carleman, T.: Sur la théorie mathématique de l'équation de Schroedinger. Arkiv Mat. Astr. o. Fys. 24, No. 11, 1–7 (1934).
Cook, J. M.: Convergence to the Møller wave-matrix. J. Math. Phys. 36, 82–87 (1957).
Flekser, M. Sh.: On the spectral function of the operator — 32-01 [in Russian]. Mat. Sbornik 38 (80), 3–22 (1956).
Gårding, L.: Eigenfunction expansions connected with elliptic differential operators. Comp. Rend. 12 Congr. Math. Scand. & Lund, 1953, pp. 44–55.
Gel'fand, I. M., & A. G. Kostyuchenko: Expansion in terms of the eigenfunctions of differential and other operators [in Russian]. Dokl. Akad. Nauk SSSR. 103, 349–352 (1955).
Hack, M. N.: On convergence to Møller wave operators. Nuovo Cimento 9, 731–733 (1958).
Hille E., & R. S. Phillips: Functional analysis and semi-groups. Amer Math. Soc. Colloq. Publ. 31 (1957).
Ito, S.: Fundamental solutions of parabolic equations and boundary value problems. Jap. J. Math. 27, 55–100 (1957).
Jauch, J. M.: Theory of the scattering operator. Helv. Phys. Acta 31, 127–158 (1958).
Jauch, J. M., & I. I. Zinnes: The asymptotic condition for simple scattering systems, Nuovo Cimento 11, 553–567 (1959).
Kato, T.: Fundamental properties of Hamiltonian operators of Schroedinger type. Trans. Amer. Math. Soc. 70, 195–211 (1951).
Kato, T.: On finite-dimensional perturbations of self-adjoint operators. J. Math. Soc. Jap. 9, 239–249 (1957).
Kato, T.: Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12, 403–425 (1959).
Kodaira, K.: The eigenvalue problem for ordinary differential equation of second order and Heisenberg's theory of S-matrices. Amer. J. Math. 71, 921–945 (1949).
Kuroda, S. T.: On the existence and the unitary property of the scattering operator, Nuovo Cimento 12, 431–454 (1959).
Mautner, F. I.: On eigenfunction expansions. Proc. Nat. Acad. Sci. U.S.A. 30, 49–53 (1953).
Miranker, W. L.: The reduced wave equations in a medium with a variable index of refraction. Comm. Pure Appl. Math. 10, 491–502 (1957).
Mott, N. F., & M. S. W. Massey: The Theory of Atomic Collisions. Oxford 1949.
Müller, C.: Grundprobleme der mathematischen Theorie elektromagnetischer Schwingungen. Grundlehren d. Math. Bd. 88. Berlin-Göttingen-Heidelberg: Springer 1957.
Müller, C.: On the behavior of the solutions of the differential equation [gDU = F(x, U) in the neighborhood of a point. Comm. Pure Appl. Math. 7, 505–516 (1954).
Møller, C.: General properties of the characteristic matrix in the theory of elementary particles, I. Det. KGL. Danske Vidensk. Selsk Mat.-Fys. Medd. 23, 2–48 (1945).
Naimark, M. A.: Linear differential operators [in Russian]. Moskva 1954.
Povzner, A. Ya.: On the expansions of arbitrary functions in terms of the eigenfunctions of the operator — [gDu + cu [in Russian]. Mat. Sbornik 32 (74), 109–156 (1953).
Riesz, F., & B. von Sz.-Nagy: Functional Analysis. New York: F. Ungar Publ. Co. 1955.
Shnol', E. E.: On the behavior of the eigenfunctions of the Schroedinger equation [in Russian]. Mat. Sbornik 42 (84), 273–286 (1957).
Stone, M. H.: Linear transformations in Hilbert space and their applications to analysis. Amer. Math. Soc. Colloq. Publ. 15 (1932).
Stummel, F.: Singuläre elliptische Differentialoperatoren in Hilbertschen Räumen. Math. Ann. 132, 150–176 (1956).
Titchmarsh, E. C.: Introduction to the Theory of Fourier Integrals. Oxford 1937.
Titchmarsh, E. C.: Eigenfunction Expansions Associated with Second-order Differential Equations, Part I. Oxford 1946.
Titchmarsh, E. C.: Eigenfunction Expansions Associated with Second-order Differential Equations, Part II. Oxford 1958.
Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörige Entwicklungen willkürlicher Funktionen. Math. Ann. 68, 220–269 (1910).
Wienholtz, E.: Halbbeschränkte partielle Differentialoperatoren zweiter Ordnung vom elliptischen Typus. Math. Ann. 135, 50–80 (1958).
Yosida, K.: On Titchmarsh-Kodaira's formula concerning Weyl-Stone's eigenfunction expansion. Nagoya Math. J. 1, 49–58 (1950).
Yosida, K.: Functional Analysis I [in Japanese]. Tokyo: Iwanami 1951.
Author information
Authors and Affiliations
Additional information
Communicated by C. Müller
Rights and permissions
About this article
Cite this article
Ikebe, T. Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory. Arch. Rational Mech. Anal. 5, 1–34 (1960). https://doi.org/10.1007/BF00252896
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00252896