Abstract
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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19 August 2019
We correct a wrong statement in the original article that the studied operator is quasi-accretive. In fact, in this corrigendum we show that the numerical range of the operator coincides with the whole complex plane. We argue that the other statements of in the original article still hold.
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Acknowledgments
The research was partially supported by the project RVO61389005 and the GACR grant No. 14-06818S.
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Kolb, M., Krejčiřík, D. Spectral analysis of the diffusion operator with random jumps from the boundary. Math. Z. 284, 877–900 (2016). https://doi.org/10.1007/s00209-016-1677-y
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DOI: https://doi.org/10.1007/s00209-016-1677-y