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Initial-boundary value problems for linear PDEs: the analyticity approach. (English) Zbl 1064.35153
Shabat, A.B.(ed.) et al., New trends in integrability and partial solvability. Proceedings of the NATO Advanced Research Workshop, Cadiz, Spain, June 12–16, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1835-5/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 132, 79-103 (2004).
Summary: It is well-known that one of the main difficulties associated with any method of solution of initial-boundary value problems for linear partial differential equations (PDEs) is due to the presence of boundary values which cannot be arbitrarily assigned. To deal efficiently with this difficulty, we have recently proposed two alternative (but interrelated) methods in Fourier space: the analyticity approach and the elimination by restriction approach.
In this work we present the analyticity approach and we illustrate its power by studying the well-posedness of initial-boundary value problems for second and third order evolutionary PDEs, and by constructing their solution. We also show the connection between the analyticity approach and the elimination by restriction approach in the particular case of the Dirichlet and Neumann problems for the Schrödinger equation in the \(n\)-dimensional quadrant.
For the entire collection see [Zbl 1050.35003].

35Q40 PDEs in connection with quantum mechanics
35Q51 Soliton equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics