We study space-inhomogeneous quantum walks (QWs) on the integer lattice which we assign three different coin matrices to positive part, negative and origin, respectively. call them two-phase QWs with one defect. They cover one-defect QWs, have been intensively researched. Localization is of most characteristic properties various types defect exhibit localization. Moreover, existence eigenvalues...

We consider Steklov eigenvalues of three-dimensional, nearly-spherical domains. In previous work, we have shown that the are analytic functions domain perturbation parameter. Here, compute first-order term asymptotic expansion, which can explicitly be written in terms Wigner 3-jsymbols. analyze expansion and prove isoperimetric result that, if l is a square integer, volume-normalized l-th eigen...

In which we show how to find the eigenvalues and eigenvectors of Cayley graphs of Abelian groups, we find tight examples for various results that we proved in earlier lectures, and, along the way, we develop the general theory of harmonic analysis which includes the Fourier transform of periodic functions of a real variable, the discrete Fourier transform of periodic functions of an integer var...

This paper studies the motion of a particle whose tune is inside and near a linear half-integer stopband. Results are found for the tune and beta functions in the stable region close to an edge of the stopband. It is shown that the eigenvalues and the eigenfunctions of the transfer matrix are real inside the stopband. All the results found are also valid for small accelerators where the large a...

We solve for spectrum, obtain explicitly and study group properties of eigenfunc-tions of Dirac operator on the Riemann sphere S 2. The eigenvalues λ are nonzero integers. The eigenfunctions are two-component spinors that belong to representations of SU (2)-group with half-integer angular momenta l = |λ| − 1 2. They form on the sphere a complete orthonormal functional set alternative to convent...

We investigate the correlation between the topological charge density and the small eigenvectors of the overlap Dirac operator on ensembles generated at four different quark masses using dynamical overlap fermions. We identify topological structures in the QCD vacuum, and by investigating their size, shape and topological charge, we determine that the small eigenvalues of the Dirac operator are...

Let A; B be two diagonal endomorphisms of the d-dimensional torus with corresponding eigenvalues relatively prime. We show that for any A-invariant ergodic measure , there exists a projection onto a torus T r of dimension r dim , that maps-almost every B-orbit to a uniformly distributed sequence in T r. As a corollary we obtain that the Hausdorr dimension of any bi-invariant measure, as well as...

Integer moments of the spectral determinant |det(zI−W )|2 of complex random matrices W are obtained in terms of the characteristic polynomial of the Hermitian matrix WW ∗ for the class of matrices W = AU where A is a given matrix and U is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results in this context...

Conditions are established under which the p-adic valuations of the invariant factors (diagonal entries of the Smith form) of an integer matrix are equal to the p-adic valuations of the eigenvalues. It is then shown that this correspondence is the typical case for “most” matrices; precise density bounds are given for when the property holds, as well as easy transformations to this typical case.

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Recently, Csikvári proved the existence of integral trees of any even diameter. In the odd case, integral trees have been constructed with diameter at most 7. In this paper, we show that for every odd integer n > 1, there are infinitely many integral trees of diameter n. AMS Mathematics Subject C...