Spectral Theory

We consider Laplacians on $\Z^2$-periodic discrete graphs. The following results are obtained: 1) The Floquet-Bloch decomposition is constructed and basic properties are derived. 2) The estimates of the Lebesgue measure of the spectrum in…

A version of the Davis-Kahan Tan $2\Theta$ theorem [SIAM J. Numer. Anal. \textbf{7} (1970), 1 -- 46] for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a recent result by…

We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in…

The difference of the resolvents of two Laplacians on a half-space subject to Robin boundary conditions is studied. In general this difference is not compact, but it will be shown that it is compact and even belongs to some…

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators.…

We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's…

A review of Finite Gap Jacobi Matrices.

The purpose of this Note is to highlight the spectral instability of some non-selfadjoint differential operators, by studying the growth rate of the norms of the spectral projections $\Pi_n$ associated with their eigenvalues. More…

For the class of unitary CMV operators with super-exponentially decaying Verblunsky coefficients we give a new proof of the inverse resonance problem of reconstructing the operator from its resonances - the zeros of the Jost function. We…

We consider the zeros on the boundary $\partial \Omega$ of a Neumann eigenfunction $\phi_{\lambda}$ of a real analytic plane domain $\Omega$. We prove that the number of its boundary zeros is $O (\lambda)$ where $-\Delta \phi_{\lambda} =…

Let ${\cal D}$ denote the class of bounded real analytic plane domains with the symmetry of an ellipse. We prove that if $\Omega_1, \Omega_2 \in {\cal D}$ and if the Dirichlet spectra coincide, $Spec(\Omega_1) = Spec(\Omega_2)$, then…

We study the spectrum of the Volterra composition operator in the space $L_2[0,1]$

We find spectrum and eigenfunctions of the operator $(Vf)(x) = \int_0^{x^\alpha} f(t) dt$ in $L_2[0,1]$

We establish a mutual relationship between main analytic objects for the dissipative extension theory of a symmetric operator $\dot A$ with deficiency indices $(1,1)$. In particular, we introduce the Weyl-Titchmarsh function $\cM$ of a…

We compute the spectrum of the "all ones" hypermatrix using the Poisson product formula. This computation includes a complete description of the eigenvalues' multiplicities, a seemingly elusive aspect of the spectral theory of tensors. We…

In this paper we investigate spectral properties of Lapla- cians on Rooms and Passages domains. In the first part, we use Dirichlet- Neumann bracketing techniques to show that for the Neumann Lapla- cian in certain Rooms and Passages…

The paper is devoted to optimization of resonances for Krein strings with total mass and statical moment constraints. The problem is to design for a given $\alpha \in \R$ a string that has a resonance on the line $\alpha + \i \R$ with a…

As an application of the Gordon lemma for orthogonal polynomials on the unit circle, we prove that for a generic set of quasiperiodic Verblunsky coefficients the corresponding two-sided CMV operator has purely singular continuous spectrum.…

This paper is devoted to the generalization of the theory of total positivity. We say that a linear operator A in R^n is generalized totally positive (GTP), if its jth exterior power preserves a proper cone K_j in the corresponding space…

A generalization of the definition of an oscillatory matrix based on the theory of cones is given in this paper. The positivity and simplicity of all the eigenvalues of a generalized oscillatory matrix are proved. The classes of generalized…