Related papers: Discrete spectrum and Weyl's asymptotic formula fo…
A trace on the C^*-algebra A of quasi-local operators on an open manifold is described, based on the results in \cite{RoeOpen}. It allows a description `a la Novikov-Shubin \cite{NS2} of the low frequency behavior of the Laplace-Beltrami…
We study in this paper the restricted roots for a class of spherical homogeneous spaces of semisimple groups which includes simply connected symmetric spaces. For these spaces we give a detailed description (case by case) of the set of…
We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some…
In this paper, by explicitly calculating the principal symbols of pseudodifferential operators and by applying H\"omander's spectral function theorem, we obtain the Weyl-type asymptotic formulas with sharp remainder estimates for the…
We consider a compact smooth manifold $X$ of dimension $n+1$ with boundary $M=\partial X$. In a collar neighborhood of $M$, we assume that the metric has the form $g=u^{-\alpha}\bar g$, where $u$ is a boundary defining function, $\alpha\in…
We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends. For simply connected ends we classify all possible cones at infinity, except for the 4-dimensional…
The study of the asymptotics of the spectral function for self-adjoint, elliptic differential, or more generally pseudodifferential, operators on a compact manifold has a long history. The seminal 1968 paper of H\"ormander, following…
We classify up to isomorphism all finite-dimensional Lie algebras that can be realised as Lie subalgebras of the complex Weyl algebra $A_1$. The list we obtain turns out to be discrete and for example, the only non-solvable Lie algebras…
We study a behavior of the conformal Laplacian operator $\L_g$ on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the…
Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and…
We develop the complex scaling for a manifold with an asymptotically cylindrical end under an assumption on the analyticity of the metric with respect to the axial coordinate of the end. We allow for arbitrarily slow convergence of the…
Under several geometric conditions imposed below, the existence of the discrete spectrum below the essential spectrum is shown for the Dirichlet Laplacian on the quantum layer built over a spherically symmetric hypersurface with a pole…
The spectrum of an arbitrary self-adjoint extension of the minimal linear relation associated with the discrete symplectic system in the limit point case is completely characterized by using the limiting Weyl--Titchmarsh…
We discuss semiclassical asymptotics for the eigenvalues of the Witten Laplacian for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by…
In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume. Surprisingly enough, when $M$ is not stochastically complete this asymptotics…
It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $\Lambda$. We show that the same is true for much wider spaces of continuous functions. In particular,…
We prove qualitative and quantitative results concerning the asymptotic density in dilates of centered convex bodies of the frequency vectors of orthogonal exponential bases and frames associated to bounded domains in Euclidean space.
In this paper, we consider the essential spectrum of submanifolds in Euclidean spaces under various geometric hypotheses. Our results involve extrinsic conditions such as finite total mean curvature, the convergence of the gradient of the…
We discuss three complementary aspects of scalar curvature singularities: asymptotic causal properties, asymptotic Ricci and Weyl curvature, and asymptotic spatial properties. We divide scalar curvature singularities into two classes:…
We provide a class of unbounded three-dimensional domains of infinite volume for which the spectrum of the associated Dirichlet Laplacian is purely discrete. The construction is based on considering tubes with asymptotically diverging…