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

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.

The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

“…For just one operator, however, there is no canonical way to choose the metric operator and the existing procedures are typically motivated by simplicity of calculation (e.g. [5][6][7]) or extra physical-like symmetries (e.g. [8,9]).…”

Multiplicities of Eigenvalues of the Diffusion Operator With Random Jumps From the Boundary

“…This research was also supported by the National Natural Science Foundation of China (Grant No. 11601372) theory and practical problems in genetics (see, for example, [1,3,4,6,7,[10][11][12][13] and the references therein). In this process, whenever the boundary of the interval [0, 1] is reached, the diffusion is redistributed in (0, 1) according to the probability distributions ν 0 , ν 1 , runs again until it hits the boundary, is redistributed and repeats this behaviour forever.…”

“…This family of non-self-adjoint differential operators has interesting spectral properties (see, for example, [3,[10][11][12][13]). In the case b 0 (x) ≡ −1, b 1 (x) ≡ 0, Leung et al [13] discovered that the whole spectrum is real despite the fact that the operator L is non-self-adjoint.…”

“…In addition, in this case the spectral gap is bounded between the lowest and the second Dirichlet eigenvalues. Recently, Kolb and Krejčiřík [10] analysed the geometric and algebraic multiplicities of the eigenvalues from a purely operator-theoretic perspective in the case…”

“…This theorem is useful for identifying the multiplicities of eigenvalues of the operator L. For example, it provides a straightforward method to obtain one of the main results in [10]…”