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

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• 作者：Martin Kolb ; David Krejčiřík
• 刊名：Mathematische Zeitschrift
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：284
• 期：3-4
• 页码：877-900
• 全文大小：646 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823
• 卷排序：284

文摘

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.