We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their L space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang [7]. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. ...

We describe the spectra and pseudospectra of continuous time and discrete time waveform relaxation operators. Since the spectrum of the finite-interval waveform relaxation operator is only a singleton, the spectrum of the infinite-interval operator is typically used to describe the behavior of waveform relaxation algorithms. Here, we show that the pseudospectrum is a more useful tool for analyz...

We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indete...

We consider the Schrödinger operator in the plane with δ-potential supported by a curve Γ. For the cases of an infinite curve and a finite loop we give estimates on the lower bound of the spectrum expressed explicitly through the strength of the interaction and a parameter c(Γ) which characterizes geometry of the curve Γ. Going further we cut the curve into finite number of pieces Γi and estima...