Ikebe, T. Eigenfunction expansions associated with the Schrödinger operators and their applications to scattering theory. (English) Zbl 0145.36902 Arch. Ration. Mech. Anal. 5, 1-34 (1960). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 139 Documents Keywords:partial differential equations PDF BibTeX XML Cite \textit{T. Ikebe}, Arch. Ration. Mech. Anal. 5, 1--34 (1960; Zbl 0145.36902) Full Text: DOI OpenURL References: [1] Berezanski, Yu. M.: Expansion in terms of the eigenfunctions of self-adjoint operators [in Russian]. Mat. Sbornik 48 (85), 75–126 (1957). [2] 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). [3] 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). · Zbl 0071.09801 [4] 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). · Zbl 0084.30702 [5] 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). [6] Carleman, T.: Sur la théorie mathématique de l’équation de Schroedinger. Arkiv Mat. Astr. o. Fys. 24, No. 11, 1–7 (1934). · JFM 60.0423.01 [7] Cook, J. M.: Convergence to the Møller wave-matrix. J. Math. Phys. 36, 82–87 (1957). [8] Flekser, M. Sh.: On the spectral function of the operator – 32-01 [in Russian]. Mat. Sbornik 38 (80), 3–22 (1956). [9] Gårding, L.: Eigenfunction expansions connected with elliptic differential operators. Comp. Rend. 12 Congr. Math. Scand. & Lund, 1953, pp. 44–55. · Zbl 0053.39101 [10] 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). [11] Hack, M. N.: On convergence to Møller wave operators. Nuovo Cimento 9, 731–733 (1958). · Zbl 0081.43401 [12] Hille E., & R. S. Phillips: Functional analysis and semi-groups. Amer Math. Soc. Colloq. Publ. 31 (1957). [13] Ito, S.: Fundamental solutions of parabolic equations and boundary value problems. Jap. J. Math. 27, 55–100 (1957). [14] Jauch, J. M.: Theory of the scattering operator. Helv. Phys. Acta 31, 127–158 (1958). · Zbl 0081.43304 [15] Jauch, J. M., & I. I. Zinnes: The asymptotic condition for simple scattering systems, Nuovo Cimento 11, 553–567 (1959). [16] Kato, T.: Fundamental properties of Hamiltonian operators of Schroedinger type. Trans. Amer. Math. Soc. 70, 195–211 (1951). · Zbl 0044.42701 [17] Kato, T.: On finite-dimensional perturbations of self-adjoint operators. J. Math. Soc. Jap. 9, 239–249 (1957). · Zbl 0089.32402 [18] Kato, T.: Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12, 403–425 (1959). · Zbl 0091.09502 [19] 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). · Zbl 0035.27101 [20] Kuroda, S. T.: On the existence and the unitary property of the scattering operator, Nuovo Cimento 12, 431–454 (1959). · Zbl 0084.44801 [21] Mautner, F. I.: On eigenfunction expansions. Proc. Nat. Acad. Sci. U.S.A. 30, 49–53 (1953). · Zbl 0050.11901 [22] Miranker, W. L.: The reduced wave equations in a medium with a variable index of refraction. Comm. Pure Appl. Math. 10, 491–502 (1957). · Zbl 0080.20902 [23] Mott, N. F., & M. S. W. Massey: The Theory of Atomic Collisions. Oxford 1949. · Zbl 0039.22401 [24] Müller, C.: Grundprobleme der mathematischen Theorie elektromagnetischer Schwingungen. Grundlehren d. Math. Bd. 88. Berlin-Göttingen-Heidelberg: Springer 1957. · Zbl 0087.21305 [25] 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). · Zbl 0056.32201 [26] 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). [27] Naimark, M. A.: Linear differential operators [in Russian]. Moskva 1954. [28] 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). [29] Riesz, F., & B. von Sz.-Nagy: Functional Analysis. New York: F. Ungar Publ. Co. 1955. [30] Shnol’, E. E.: On the behavior of the eigenfunctions of the Schroedinger equation [in Russian]. Mat. Sbornik 42 (84), 273–286 (1957). · Zbl 0078.27904 [31] Stone, M. H.: Linear transformations in Hilbert space and their applications to analysis. Amer. Math. Soc. Colloq. Publ. 15 (1932). · Zbl 0005.40003 [32] Stummel, F.: Singuläre elliptische Differentialoperatoren in Hilbertschen Räumen. Math. Ann. 132, 150–176 (1956). · Zbl 0070.34603 [33] Titchmarsh, E. C.: Introduction to the Theory of Fourier Integrals. Oxford 1937. · Zbl 0017.40404 [34] Titchmarsh, E. C.: Eigenfunction Expansions Associated with Second-order Differential Equations, Part I. Oxford 1946. · Zbl 0061.13505 [35] Titchmarsh, E. C.: Eigenfunction Expansions Associated with Second-order Differential Equations, Part II. Oxford 1958. · Zbl 0097.27601 [36] Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörige Entwicklungen willkürlicher Funktionen. Math. Ann. 68, 220–269 (1910). · JFM 41.0343.01 [37] Wienholtz, E.: Halbbeschränkte partielle Differentialoperatoren zweiter Ordnung vom elliptischen Typus. Math. Ann. 135, 50–80 (1958). · Zbl 0142.37701 [38] Yosida, K.: On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eigenfunction expansion. Nagoya Math. J. 1, 49–58 (1950). · Zbl 0038.24802 [39] Yosida, K.: Functional Analysis I [in Japanese]. Tokyo: Iwanami 1951. · Zbl 0045.21201 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.