Related papers: Little Magnetic Book
For a two-dimensional curved waveguide, it is well known that the spectrum of the Dirichlet Laplacian is unstable. Any perturbation of the straight strip produces eigenvalues below the essential spectrum. In this paper, a magnetic field is…
We consider the Hamiltonian for a charged particle in a harmonic potential in the presence of a magnetic field. The most symmetric case depends on one parameter, the variation of which leads from a spectrum bounded from below to an…
This paper is dedicated to the spectral analysis of the semiclassical purely magnetic Laplacian on the plane in the situation where the magnetic field $B$ vanishes nondegenerately on an open smooth curve $\Gamma$. We prove the existence of…
In this paper we investigate a spectra of the Laplacian matrix of cyclic groups using the properties of their characteristic polynomials. We have proved several assertions about the relationship between the spectra of different groups.
We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the…
In this paper we study the coisotropic reduction in different types of dynamics according to the geometry of the corresponding phase space. The relevance of the coisotropic reduction is motivated by the fact that these dynamics can always…
Two non-local asymptotic invariants of magnetic fields for the ideal magnetohydrodynamics are introduced. The velocity of variation of the invariants for a non-ideal magnetohydrodynamics with a small magnetic dissipation is estimated. By…
This article explores some geometric and algebraic properties of the dynamical system which is represented by matrix differential equations arising from inertial navigation problems, such as the symplecticity and the orthogonality.…
This mini-course of 20 lectures aims at highlights of spectral theory for self-adjoint partial differential operators, with a heavy emphasis on problems with discrete spectrum. Part I: Discrete Spectrum (ODE preview, Laplacian - computable…
We consider a "convolution mm-Laplacian" operator on metric-measure spaces and study its spectral properties. The definition is based on averaging over small metric balls. For reasonably nice metric-measure spaces we prove stability of…
Motivated by research on contraction analysis and incremental stability/stabilizability the study of 'differential properties' has attracted increasing attention lately. Previously lifts of functions and vector fields to the tangent bundle…
We investigate several aspects of the nodal geometry and topology of Laplace eigenfunctions, with particular emphasis on the low frequency regime. This includes investigations in and around the Payne property, opening angle estimates of…
Some geometric structures with associated Riemannian metrics have been considered in the book.
On a closed, orientable Riemannian surface $\Sigma_g$ of arbitrary genus $g\geq 1$ and Riemannian metric $h$ we study the magnetic Laplacian with magnetic potential given by a harmonic $1$-form $A$. Its lowest eigenvalue (magnetic ground…
Dynamical correlation functions of the toric code in a uniform magnetic field are studied inside the topological phase, in the small-field limit. Such an experimentally measurable quantity displays rich field-dependent features that can be…
We consider a negative Laplacian in multi-dimensional Euclidean space (or a multi-dimensional layer) with a weak disorder random perturbation. The perturbation consists of a sum of lattice translates of a delta interaction supported on a…
A monograph on the mathematical aspects of Special Relativity, focusing on the Lorentz group and the properties of relativistic transformations in mechanics and electrodynamics. Manuscript of published book, with added appendices.
We consider the Dirichlet Laplacian in a family of narrow unbounded domains. As the width of these domains goes to 0, we study the asymptotic behavior of the eigenvalues that lie below the essential spectrum and the asymptotic behavior of…
We study the 3D Neuman magnetic Laplacian in the presence of a semi-classical parameter and a non-uniform magnetic field with constant intensity. We determine a sharp two term asymptotics for the lowest eigenvalue, where the second term…
We numerically construct the spectrum of the Laplacian on Page's inhomogeneous Einstein metric on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ by reducing the problem to a (singular) Sturm-Liouville problem in one dimension. We perform a…