Topological states of matter can be classified only in terms global topological invariants. These invariants are encoded observable phase factors the state vectors electrons. In condensed matter, energy spectrum Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The each continuous piece depends on quasi-momentum which varies Brillouin zone. Thus, zone variables...

Consider a family of bounded domains Ωt in the plane (or more generally any Euclidean space) that depend analytically on the parameter t, and consider the ordinary Neumann Laplacian ∆t on each of them. Then we can organize all the eigenfunctions into continuous families u (j) t with eigenvalues λ (j) t also varying continuously with t, although the relative sizes of the eigenvalues will change ...

In previous studies we first concentrated on utilizing crisp simulationto produce discrete event fuzzy systems simulations. Then we extendedthis research to the simulation of continuous fuzzy systems models. In this paperwe continue our study of continuous fuzzy systems using crisp continuoussimulation. Consider a crisp continuous system whose evolution depends ondifferential equations. Such a ...

Shape analysis plays a pivotal role in a large number of applications, ranging from traditional geometry processing to more recent 3D content management. In this scenario, spectral methods are extremely promising as they provide a natural library of tools for shape analysis, intrinsically defined by the shape itself. In particular, the eigenfunctions of the Laplace-Beltrami operator yield a set...

We consider a U(1)-invariant nonlinear Dirac equation, interacting with itself via the mean field mechanism. We analyze the longtime asymptotics of solutions and prove that, under certain generic assumptions, each finite charge solution converges as t → ±∞ to the twodimensional set of all “nonlinear eigenfunctions” of the form φ(x)e−iωt . This global attraction is caused by the nonlinear energy...

We solve the Regge-Wheeler equation for axial perturbations of the Schwarzschild metric in the black hole interior in terms of Heun’s functions and give a description of the spectrum and the eigenfunctions of the interior problem. The phenomenon of attraction and repulsion of the discrete eigenvalues of gravitational waves is discovered.

We study the spectrum of a one-dimensional Schrödinger operator perturbed by a fast oscillating potential. The oscillation period is a small parameter. The essential spectrum is found in an explicit form. The existence and multiplicity of the discrete spectrum are studied. The complete asymptotics expansions for the eigenvalues and the associated eigenfunctions are constructed.

The quantum modes of a new family of relativistic oscillators are studied by using the supersymmetry and shape invariance in a version suitable for (1+1) dimensional relativistic systems. In this way one obtains the Rodrigues formulas of the normalized energy eigenfunctions of the discrete spectra and the corresponding rising and lowering operators. Pacs: 04.62.+v, 03.65.Ge

We prove the Anderson localization near the bottom of the spectrum for two dimensional discrete Schrödinger operators with a class of random vector potentials and no scalar potentials. Main lemmas are the Lifshitz tail and the Wegner estimate on the integrated density of states. Then, the Anderson localization, i.e., the pure point spectrum with exponentially decreasing eigenfunctions, is prove...