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We consider a generalization of the diffusion equation on graphs. This generalized diffusion equation gives rise to both normal and superdiffusive processes on infinite one-dimensional graphs. The generalization is based on the $k$-path Laplacian operators $L_{k}$, which account for the hop of a diffusive particle to non-nearest neighbours in a graph. We first prove that the $k$-path Laplacian operators are self-adjoint. Then, we study the transformed $k$-path Laplacian operators using Laplace, factorial and Mellin transforms. We prove that the generalized diffusion equation using the Laplace- and factorial-transformed operators always produce normal diffusive processes independently of the parameters of the transforms. More importantly, the generalized diffusion equation using the Mellin-transformed $k$-path Laplacians $sum_{k=1}^{infty}k^{-s}L_{k}$ produces superdiffusive processes when $1<s<3$.
We show that the deficiency indices of the minimal Gaffney Laplacian on an infinite locally finite metric graph are equal to the number of finite volume graph ends. Moreover, we provide criteria, formulated in terms of finite volume graph ends, for the Gaffney Laplacian to be closed.
We revisit an archive submission by P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, arXiv:1908.03795, on $n times n$ self-adjoint matrices from the point of view of self-adjoint Dirichlet Schrodinger operators on a compact interval.
We identify explicitly the fractional power spaces for the $L^2$ Dirichlet Laplacian and Dirichlet Stokes operators using the theory of real interpolation. The results are not new, but we hope that our arguments are relatively accessible.
We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided Levy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelli
In this paper we investigate the one-dimensional harmonic oscillator with a singular perturbation concentrated in one point. We describe all possible selfadjoint realizations and we show that for certain conditions on the perturbation exactly one neg