具有边界随机跳的扩散算子的谱分析
Spectral analysis of the diffusion operator with random jumps from the boundary
摘要：
使用运营商的理论框架中的希尔伯特空间设置，我们进行了详细的谱分析的一维拉普拉斯算子在有界的间隔，受到特定的非自伴连接的边界条件建模的随机跳跃从边界到一个点内的间隔。根据以前的工作，我们发现所有的特征值是真实的。作为新的结果，我们导出并分析了伴随算子，确定了本征值的几何重数和代数重数，给出了本征函数和广义本征函数的计算公式，并研究了它们的基性质.事实证明，后者在很大程度上取决于内点到区间中心的距离除以区间的长度是有理数还是无理数。最后，我们找到了一个封闭的度量算子，提供了一个相似性变换的问题，一个自伴算子的公式。
英文摘要：
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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