The Moore-Penrose pseudo-inverse $X^\dagger$, defined for rectangular matrices, naturally emerges in many areas of mathematics and science. For a pair of rectangular matrices $X, Y$ where the corresponding entries are jointly Gaussian and i.i.d., we analyse the support of the eigenvalue spectrum of $XY^\dagger$.
We provide in this Letter a two-point generalisation of the Agmon estimate for Schrödinger operators on graphs recently established by S. Steinerberger. It reduces to his estimate when the two points belong to different sets separated by the potential and the energy, i.e., the allowed and forbidden regions.
In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schrödinger operators with non-negative bounded potentials and subject to Neumann boundary conditions.
We find the spectrum and eigenvectors of an arbitrary irreducible complex tridiagonal matrix with two-periodic main diagonal provided that the spectrum and eigenvectors of the matrix with the same sub- and superdiagonals and zero main diagonal is known. Our result substantially generalises some recent results on the Sylvester-Kac matrix and its certain main principal submatrices.
We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb--Thirring inequality due to Hansmann--Katriel. The spectral enclosure for such operators is also discussed.
We study the eigenvalues of the Dirichlet-to-Neumann operator on a finite subgraph of the integer lattice Zn. We estimate the first n+1 eigenvalues using the number of vertices of the subgraph. As a corollary, we prove that the first non-trivial eigenvalue of the Dirichlet-to-Neumann operator tends to zero as the number of vertices of the subgraph tends to infinity.
This memoir is devoted to a part of the results from the author about two topics: in the first part, the asymptotics of the low-lying eigenvalues of Schrödinger operators in domains that may have corners, and in the second part, the analysis of the thresholds of a class of fibered operators. The main common object is the magnetic Laplacian, and the two parts are connected through the study of model problems in unbounded domains.
Alexander Nazarov, Sergey Nazarov, German Zavorokhin
The existence of a symmetric mode in an elastic solid wedge for all admissible values of the Poisson ratio and arbitrary openings close to $π$ has been proven.
Abdelaziz Tajmouati, Abdeslam El Bakkali, M. B. Mohamed Ahmed
In this paper, we study the operator equation $AB=λBA$ for a bounded operator $A,B$ on a complex Hilbert space. We focus on algebraic relations between different operators that include normal, $M$-hyponormal, quasi $*$-paranormal and other classes.
We consider two-dimensional Schroedinger operators with magnetic field. Under certain regularity and decay assumptions on the magnetic and electric field we show that the behavior of the corresponding resolvent at threshold and the behavior of the unitary group for large times are determined by the total flux of the associated magnetic field.
This papers deals with congruence subgroups of convex cocompact subgroups of PSL2(Z). We examine the behaviour of the resonance spectrum when the congruence parameter q goes to infinity: we show a lower bound for the counting function in discs and an upper bound in vertical strips. These results show drastically different behaviour on both sides of the critical line $\Re(s)=δ/2$.
We consider Schrödinger operators on periodic discrete graphs. It is known that the spectrum of these operators has band structure. We obtain a localization of spectral bands in terms of eigenvalues of Dirichlet and Neumann operators on a finite graph, which is constructed from the fundamental cell of the periodic graph. The proof is based on the Floquet decomposition of Schrödinger operators and the minimax principle.
In this paper, we study spectral properties of generalized weighted Hilbert matrices. In particular, we establish results on the spectral norm, determinant, as well as various relations between the eigenvalues and eigenvectors of such matrices. We also study the asymptotic behaviour of the spectral norm of the classical Hilbert matrix.
We study sub-Dirac operators that are associated with left-invariant bracket-generating sub-Riemannian structures on compact quotients of nilpotent semi-direct products $G=\mathbb{R}^n\rtimes_A\mathbb{R}$. We will prove that these operators admit an $L^2$-basis of eigenfunctions. Explicit examples show that the spectrum of these operators can be non-discrete and that eigenvalues may have infinite multiplicity.
The scattering matrix of the Schrodinger operator with smooth short-range electric and magnetic potentials is considered. The asymptotic density of the eigenvalues of this scattering matrix in the high energy regime is determined. An explicit formula for this density is given. This formula involves only the magnetic vector-potential.
We introduce the spectral points of two-sided positive type of bounded normal operators in Krein spaces. It is shown that a normal operator has a local spectral function on sets which are of two-sided positive type. In addition, we prove that the Riesz-Dunford spectral subspace corresponding to a spectral set which is only of positive type is uniformly positive. The restriction of the operator to this subspace is then normal in a Hilbert space.
We prove spectral and dynamical localization for Anderson models on locally finite graphs using the fractional moment method. Our theorems extend earlier results on localization for the Anderson model on $\ZZ^d$. We establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder.