References
Lax, P.D., & R.S. Phillips, Scattering theory. Bull. Amer. Math. Soc. 70, 130–142 (1964).
Shizuta, Y., Eigenfunction expansion associated with the operator — Δ in the exterior domain. Proc. Japan Acad. 39, 656–660 (1963).
Ikebe, T., Orthogonality of the eigenfunctions for the exterior problem connected with — Δ. Arch. Rational Mech. Anal. 19, 71–73 (1965).
Werner, P., Randwertprobleme der mathematischen Akustik. Arch. Rational Mech. Anal. 10, 29–66 (1962).
Ikebe, T., Eigenfunction expansions associated with the Schroedinger operator and their applications to scattering theory. Arch. Rational Mech. Anal. 5, 1–34 (1960).
Thoe, D., Scattering theory for the wave equation with a potential. Doctoral Thesis, Stanford University (1965), 147–155.
Hörmander, L., Linear Partial Differential Operators, p. 1–56. New York: Academic Press 1963.
Nirenberg, L., Remarks on strongly elliptic partial differential equations. Comm. Pure Appl. Math. 8, 649–675 (1955).
Schechter, M., General boundary value problems for elliptic partial differential equations. Comm. Pure Appl. Math. 12, 457–486 (1959).
Gårding, L., Dirichlet's problem for linear elliptic partial differential equations. Math. Scand. 1, 55–72 (1953).
Watson, G., The Theory of Bessel Functions, p. 73. Cambridge: University Press 1922.
Miranda, C., Equazioni alle Derivate Parziali di Tipo Ellittico, p. 1–35. Berlin-Göttingen-Heidelberg: Springer 1955.
Stone, M., Linear Transformations in Hilbert Space, p. 183. New York: American Mathematical Society 1932.
Titchmarsh, E., Introduction to the Theory of Fourier Integrals, p. 31. London: Oxford University Press 1937.
Bers, L., F. John, & M. Schechter, Partial Differential Equations, p. 221. New York: Interscience 1964.
John, F., Plane Waves and Spherical Means, p. 36. New York: Interscience 1955.
Author information
Authors and Affiliations
Additional information
Communicated by M. M. Schiffer
Most of this material was contained in the author's dissertation, written at Stanford University. The research was supported in part by a U.S. National Science Foundation Cooperative Summer Fellowship and U.S. National Science Foundation Grant 1883.
Rights and permissions
About this article
Cite this article
Shenk, N.A. Eigenfunction expansions and scattering theory for the wave equation in an exterior region. Arch. Rational Mech. Anal. 21, 120–150 (1966). https://doi.org/10.1007/BF00266571
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00266571