Related papers: Weyl's laws and Connes' Trace Theorem for operator…

Schroedinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner-von Neumann type is considered. The asymptotics of generalized eigenvectors for the values of the spectral parameter from…

Using {\it weighted traces} which are linear functionals of the type $$A\to tr^Q(A):=(tr(A Q^{-z})-z^{-1} tr(A Q^{-z}))_{z=0}$$ defined on the whole algebra of (classical) pseudo-differential operators (P.D.O.s) and where $Q$ is some…

We give a geometric proof of a theorem of Weyl on the continuous part of the spectrum of Sturm-Liouville operators on the half-line with asymptotically constant coefficients. Earlier proofs due to Weyl and Kodaira depend on special features…

In this paper, we consider the trace property of pseudo-differential operators with symbols in $\alpha$-modulation spaces.

Let s 1 ,. .. , s k be the elementary symmetric functions of the complex variables x 1 ,. .. , x k. We say that F $\in$ C[s 1 ,. .. , s k ] is a trace function if their exists f $\in$ C[z] such that F (s 1 ,. .. , s k ] = k j=1 f (x j) for…

We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the non-selfadjoint operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We…

We prove a sharp Weyl estimate for the number of eigenvalues belonging to a fixed interval of energy of a self-adjoint difference operator acting on $\ell^2(\epsilon\mathbb{Z}^d)$ if the associated symplectic volume of phase space in…

This work is concerned with extending the results of Calder\' on and Vaillancourt proving the boundedness of Weyl pseudo differential operators Op_h^{weyl} (F) in L^2(\R^n). We state conditions under which the norm of such operators has an…

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We…

We prove a two-term Weyl-type asymptotic law, with error term O(1/n), for the eigenvalues of the operator psi(-Delta) in an interval, with zero exterior condition, for complete Bernstein functions psi such that x psi'(x) converges to…

We extend the isospectral deformations of Connes, Landi and Dubois-Violette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group $R^l$. Under deformation by a torus action, a standard formula…

We consider discrete Schroedinger operator J with Wigner-von Neumann potential not belonging to l^2. We find asymptotics of orthonormal polynomials associated to J. We prove the Weyl-Titchmarsh type formula, which relates the spectral…

We deal with the asymptotic behaviour for $\lambda\to+\infty$ of the counting function $N_P(\lambda)$ of certain positive selfadjoint operators $P$ with double order $(m,\mu)$, $m,\mu>0$, $m\not=\mu$, defined on a manifold with ends $M$.…

We prove equivariant spectral asymptotics for $ h$-pseudodifferential operators for compact orthogonal group actions generalizing results of El-Houakmi and Helffer (1991) and Cassanas (2006). Using recent results for certain oscillatory…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V$ with the radial potential $V(x)=F(|x|), F(r)\ge 0$, we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues, as the coupling parameter…

We deduce eigenvalue asymptotics of the Neumann--Poincar\'e operators in three dimensions. The region $\Omega$ is $C^{2, \alpha}$ ($\alpha>0$) bounded in ${\mathbf R}^3$ and the Neumann--Poincar\'e operator ${\mathcal K}_{\partial\Omega} :…

Small perturbations of the Jacobi matrix with weights \sqrt n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical…

The spectral properties of non-self-adjoint extensions $A_{[B]}$ of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in…

The Wodzicki residue is the unique trace on the algebra of classical pseudodifferential operators on a closed manifold, and Connes in 1988 proved that it coincides with the Dixmier trace. A Carnot manifold is a manifold $M$ whose tangent…

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and…