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

M. Kolb, D. Krejčiřík, Mathematische Zeitschrift 284 (2016) 877–900.

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Journal Article | Published | English
Author
Kolb, MartinLibreCat; Krejčiřík, David
Department
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.
Publishing Year
Journal Title
Mathematische Zeitschrift
Volume
284
Page
877-900
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Kolb M, Krejčiřík D. Spectral analysis of the diffusion operator with random jumps from the boundary. Mathematische Zeitschrift. 2016;284:877-900. doi:https://link.springer.com/content/pdf/10.1007/s00209-016-1677-y.pdf
Kolb, M., & Krejčiřík, D. (2016). Spectral analysis of the diffusion operator with random jumps from the boundary. Mathematische Zeitschrift, 284, 877–900. https://link.springer.com/content/pdf/10.1007/s00209-016-1677-y.pdf
@article{Kolb_Krejčiřík_2016, title={Spectral analysis of the diffusion operator with random jumps from the boundary}, volume={284}, DOI={https://link.springer.com/content/pdf/10.1007/s00209-016-1677-y.pdf}, journal={Mathematische Zeitschrift}, publisher={Springer}, author={Kolb, Martin and Krejčiřík, David}, year={2016}, pages={877–900} }
Kolb, Martin, and David Krejčiřík. “Spectral Analysis of the Diffusion Operator with Random Jumps from the Boundary.” Mathematische Zeitschrift 284 (2016): 877–900. https://link.springer.com/content/pdf/10.1007/s00209-016-1677-y.pdf.
M. Kolb and D. Krejčiřík, “Spectral analysis of the diffusion operator with random jumps from the boundary,” Mathematische Zeitschrift, vol. 284, pp. 877–900, 2016, doi: https://link.springer.com/content/pdf/10.1007/s00209-016-1677-y.pdf.
Kolb, Martin, and David Krejčiřík. “Spectral Analysis of the Diffusion Operator with Random Jumps from the Boundary.” Mathematische Zeitschrift, vol. 284, Springer, 2016, pp. 877–900, doi:https://link.springer.com/content/pdf/10.1007/s00209-016-1677-y.pdf.

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