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Schrödinger equation with noise on the boundary

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  • We treat the question of existence and uniqueness of distributional solutions for the linear Schrödinger equation in a bounded domain with boundary noise. We cover both Dirichlet and Neumann noise. For the proof we make use of spectral decomposition of the Laplacian with homogeneous Neumann/Direchlet boundary condition.
    Mathematics Subject Classification: Primary: 60H15; Secondary: 35G16, 35Q41.

    Citation:

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  • [1]

    H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992)" (eds. H.-J. Schmeisser), Teubner-Texte Math., 133, Teubner, Stuttgart (1993), 9-126.

    [2]

    G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary conditions, Stochastics Stochastics Rep. 42 (1993), 167-182.

    [3]

    I. Lasiecka and R. Triggiani, "Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory" Lecture Notes in Control and Information Sciences, 164. Springer-Verlag, Berlin, 1991.

    [4]

    J.-L. Lions and E. Magenes, "Non-homogeneous boundary value problems and applications. Vol. I.," Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

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