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Item type:Publication, Access status: with files , Minimal Families of Limit Operators(2022-06)We study two abstract scenarios, where an operator family has a certain minimality property. In both scenarios, it is shown that norm, spectrum and resolvent are the same for all family members. Both abstract settings are illustrated by practically relevant examples, including discrete Schrödinger operators with periodic, quasiperiodic, almost-periodic, Sturmian and pseudo-ergodic potential. The main tool is the method of limit operators, known from studies of Fredholm operators and convergence of projection methods. We close by connecting this tool to the study of subwords of the operator potential.Publicationtype: Journal Article159 19 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Dirichlet forms for singular diffusion on graphsWe describe operators driving the time evolution of singular diffusion on finite graphs whose vertices are allowed to carry masses. The operators are defined by the method of quadratic forms on suitable Hilbert spaces. The model also covers quantum graphs and discrete Laplace operators.Publicationtype: Journal Article54 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator(2010-03-20); ; The purpose of this paper is to prove that the spectrum of the non-self-adjoint one-particle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433--6443) has interior points. We do this by first recalling that the spectrum of this random operator is the union of the set of ℓ∞ eigenvalues of all infinite matrices with the same structure. We then construct an infinite matrix of this structure for which every point of the open unit disk is an ℓ∞ eigenvalue, this following from the fact that the components of the eigenvector are polynomials in the spectral parameter whose non-zero coefficients are ± 1's, forming the pattern of an infinite discrete Sierpinski triangle.Publicationtype: Journal Article44 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Fredholmness and index of operators in the Wiener algebra are independent of the underlying space(2008)The purpose of this paper is to demonstrate the so-called Fredholm-inverse closedness of the Wiener algebra W and to deduce independence of the Fredholm property and index of the underlying space. More precisely, we look at operators A ∈ W as acting on a family of vector valued p spaces and show that the Fredholm regularizer of A for one of these spaces can always be chosen in W as well and therefore regularizes A (modulo compact operators) on all of the p spaces under consideration. We conclude that both Fredholmness and the index of A do not depend on the p space that A is considered as acting on.Publicationtype: Journal Article30