Integration of the equation of evolution in a Banach space
01 Jul 1953-Journal of The Mathematical Society of Japan (The Mathematical Society of Japan)-Vol. 5, Iss: 2, pp 208-234
About: This article is published in Journal of The Mathematical Society of Japan.The article was published on 1953-07-01 and is currently open access. It has received 285 citations till now. The article focuses on the topics: Infinite-dimensional vector function & Banach manifold.
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01 Jan 1975
843 citations
Cites methods from "Integration of the equation of evol..."
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455 citations
TL;DR: In this article, O'Connor's approach to spatial exponential decay of eigenfunctions for multiparticle Schrodinger Hamiltonians is developed from the point of view of analytic perturbations with respect to transformation groups.
Abstract: O'Connor's approach to spatial exponential decay of eigenfunctions for multiparticle Schrodinger Hamiltonians is developed from the point of view of analytic perturbations with respect to transformation groups. This framework allows an improvement of his results in some directions; in particular if interactions are dilation analytic, exponential fall-off is shown to hold for any bound-state wave-function corresponding to an eigenvalue distinct from thresholds; it is shown that the exponential decay rate depends on the distance from the bound-state energy to the nearest threshold. Applications include non existence of positive energy bound-states.
390 citations
TL;DR: In this paper, the authors considered the uniqueness of the solutions of the basic boundary value problems by Rothe's method and the Green's function for the solution of the first boundary value problem.
Abstract: CONTENTSIntroduction § 1. The maximum principle. Uniqueness of the solutions of the basic boundary value problems § 2. A priori estimates § 3. Solution of boundary value problems by Rothe's method. The Cauchy problem § 4. The fundamental solution of a linear parabolic equation. The Green's function. The method of integral equations for the solution of boundary value problems § 5. Generalized solutions of boundary value problems. The uniqueness theorem. Some auxiliary propositions § 6. The method of finite differences § 7. Some methods of functional analysis for the solution of boundary value problems § 8. The solution of boundary value problems by the method of continuation by a parameter § 9. The application of Galerkin's method for the construction of a solution of the first boundary value problem § 10. Generalized solutions of Cauchy's problem § 11. On differentiability properties of generalized solutions § 12. The behaviour of solutions for indefinitely increasing timeReferences
294 citations
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Book•
01 Jan 1948
TL;DR: The theory of semi-groups has been studied extensively in the literature, see as discussed by the authors for a survey of some of the main applications of semi groups in the context of functional analysis.
Abstract: Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: Subadditive functions Semi-modules Addition theorem in a Banach algebra Semi-groups in the strong topology Generator and resolvent Generation of semi-groups Part Three. Advanced Analytical Theory of Semi-Groups: Perturbation theory Adjoint theory Operational calculus Spectral theory Holomorphic semi-groups Applications to ergodic theory Part Four. Special Semi-groups and Applications: Translations and powers Trigonometric semi-groups Semi-groups in $L_p(-\infty,\infty)$ Semi-groups in Hilbert space Miscellaneous applications Part Five. Extensions of the theory: Notes on Banach algebras Lie semi-groups Functions on vectors to vectors Bibliography Index.
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