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Spectral analysis of the diffusion operator with random jumps from the boundary

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    0466585 - ÚJF 2017 RIV DE eng J - Journal Article
    Kolb, M. - Krejčiřík, David
    Spectral analysis of the diffusion operator with random jumps from the boundary.
    Mathematische Zeitschrift. Roč. 284, 3-4 (2016), s. 877-900. ISSN 0025-5874
    R&D Projects: GA ČR(CZ) GA14-06818S
    Institutional support: RVO:61389005
    Keywords : self-adjoint operators * eigenvalues * eigenfunctions
    Subject RIV: BE - Theoretical Physics
    Impact factor: 0.738, year: 2016

    Using an operator-theoretic framework in a Hilbert-space setting, we perform a detailed spectral analysis of the one-dimensional Laplacian in a bounded interval, subject to specific non-self-adjoint connected boundary conditions modelling a random jump from the boundary to a point inside the interval. In accordance with previous works, we find that all the eigenvalues are real. As the new results, we derive and analyse the adjoint operator, determine the geometric and algebraic multiplicities of the eigenvalues, write down formulae for the eigenfunctions together with the generalised eigenfunctions and study their basis properties. It turns out that the latter heavily depend on whether the distance of the interior point to the centre of the interval divided by the length of the interval is rational or irrational. Finally, we find a closed formula for the metric operator that provides a similarity transform of the problem to a self-adjoint operator.
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