Related papers: Probabilistic Weyl laws for quantized tori
In this article, we study the convergence of the empirical spectral measure of twisted Toeplitz matrices subject to small random perturbations. We show that the empirical spectral measure converges weakly in probability to the push-forward…
We study a toy model for "partially open" wave-mechanical system, like for instance a dielectric micro-cavity, in the semiclassical limit where ray dynamics is applicable. Our model is a quantized map on the 2-dimensional torus, with an…
We develop a calculus of Berezin-Toeplitz operators quantizing exotic classes of smooth functions on compact K\"ahler manifolds and acting on holomorphic sections of powers of positive line bundles. These functions (classical observables)…
The transmission eigenvalue problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. In this work, we establish the Weyl law for the eigenvalues and the completeness of the…
The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation…
In this work we consider semi-classical Schr\"odinger operators with potentials supported in a bounded strictly convex subset ${\cal O}$ of ${\bf R}^n$ with smooth boundary. Letting $h$ denote the semi-classical parameter, we consider…
In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in…
The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this…
The Weyl law of the Laplacian on the flat torus $\mathbb{T}^n$ is concerning the number of eigenvalues $\le\lambda^2$, which is equivalent to counting the lattice points inside the ball of radius $\lambda$ in $\mathbb{R}^n$. The leading…
In this paper, we give a correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the torus $ \mathbb{T}^2$. To achieve this, we use the correspondence between the Berezin-Toeplitz and the complex Weyl quantizations…
In this paper, we consider elliptic differential operators on compact manifolds with a random perturbation in the 0th order term and show under fairly weak additional assumptions that the large eigenvalues almost surely distribute according…
We consider Berezin-Toeplitz operators on compact Kahler manifolds whose symbols are characteristic functions. When the support of the characteristic function has a smooth boundary, we prove a two-term Weyl law, the second term being…
We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.
We find the Weyl law followed by the eigenvalues of contractive maps. An important property is that it is mainly insensitive to the dimension of the corresponding invariant classical set, the strange attractor. The usual explanation for the…
Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional quasi-relativistic Hamiltonian (-h^2 c^2 d^2/dx^2 + m^2 c^4)^(1/2) + V_well(x) (the Klein-Gordon square-root operator with electrostatic potential) with the infinite…
An exact semiclassical version of the classical KAM theorem about small perturbations of vector fields on the torus is given. Moreover, a renormalization theorem based on counterterms for some semiclassical systems that are close to being…
The purpose of this paper is to review the asymptotic distribution of eigenvalues of the Dirichlet Laplacian. We introduce and recall all the relevant spectral quantities and provide a proof based on the Fourier Tauberian Theorem.
We show that Weyl's law for the number and the Riesz means of negative eigenvalues of Schr\"odinger operators remains valid under minimal assumptions on the potential, the vector potential and the underlying domain.
Quantum mechanics, one of the keystones of modern physics, exhibits several peculiar properties, differentiating it from classical mechanics. One of the most intriguing is that variables might not have definite values. A complete quantum…
We consider a class of one-dimensional nonselfadjoint semiclassical pseudo-differential operators, subject to small random perturbations, and study the statistical properties of their (discrete) spectra, in the semiclassical limit $h\to 0$.…