We consider a Dirichlet spectral problem for second order differential operator, with piecewise constant coefficients, in domain ?? the plane R2. Here is ??????, where ? fixed bounded boundary ?, ?? curvilinear band of width O(?), and ?=??????. The density stiffness constants are ??m?t ??t respectively this band, while they 1 ?; t?1, m>2, ? small positive parameter. address asymptotic behavior,...

Abstract Mathematical analysis on the existence of eigenvalues is essential because it deeply related to localization, which an exceptionally crucial property quantum walks (QWs). We construct method for eigenvalue problem via transfer matrix space-inhomogeneous three-state QWs in one dimension with a self-loop, extension technique previous study (Kiumi and Saito 2021 Quantum Inf. Process. 20 1...

We establish a moderate deviation principle (MDP) for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on fine asymptotics of the variance of the eigenvalue counting function of GUEmatrices due to Gustavsson. The extension to certain families of Wigner matrices is based on the Tao and Vu Four Moment Theorem and applies localization results by Erdös, Yau and Yin. Mor...

We consider a gyroscopic system with two degrees of freedom under the action of small dissipative and non-conservative positional forces, which has its origin in the models of rotating bodies of revolution being in frictional contact. The spectrum of the unperturbed gyroscopic system forms a ”spectral mesh” in the plane ”frequency – gyroscopic parameter” with double semi-simple purely imaginary...

The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct aymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by ...

In the Anderson model on Zd, we consider a sequence of its finite volume approximation Hk k and construct a set of sequences composed of the eigenvalues and eigenfunctions of Hk in the localized region I which converge to those of H simultaneously. For its proof, Minami’s estimate turns out to be important. This result implies that, in the localized region, each eigenfunction behaves almost ind...

We consider a weakly interacting quantum spin chain with random local interactions. We prove that many-body localization follows from a physically reasonable assumption that limits the extent of level attraction in the statistics of eigenvalues. In a Kolmogorov-Arnold-Moser-style construction, a sequence of local unitary transformations is used to diagonalize the Hamiltonian by deforming the in...

Abstract. We study localization and derive stochastic estimates (in particular, Wegner and Minami estimates) for the eigenvalues of weakly correlated random discrete Schrödinger operators in the localized phase. We apply these results to obtain spectral statistics for general discrete alloy type models where the single site perturbation is neither of finite rank nor of fixed sign. In particular...