Related papers: Eigenvalue distributions and Weyl laws for semi-cl…
We consider quite general $h$-pseudodifferential operators on $R^n$ with small random perturbations and show that in the limit of small $h$ the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The…
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…
We study the distribution of eigenvalues for non-selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two, assuming that the classical flow of the unperturbed part is completely…
We consider quite general differential operators on the circle with a small random lower order perturbation. We embrace two points a view, the semiclassical and the high energy limits. We show (a) in the semiclassical limit, that the…
In this work we extend a previous work about the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint differential operators with small multiplicative random perturbations, by treating the case of operators on compact…
We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schr\"odinger operators in the presence of two potential wells and with an energy-level crossing. We provide…
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 $h\to 0$.…
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 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…
We study the distribution of eigenvalues for selfadjoint $h$--pseudodifferential operators in dimension two, arising as perturbations of selfadjoint operators with a periodic classical flow. When the strength $\varepsilon$ of the…
We consider a string with fixed endpoints where the mass density and/or the elastic coefficient vary in a self-affine way as function of position. It is demonstrated how the eigenvalues in the asymptotic limit are distributed. Scaling laws…
We develop a theory which describes the behaviour of eigenvalues of a class of one-dimensional random non-Hermitian operators introduced recently by Hatano and Nelson. Under general assumptions on random parameters we prove that the…
We consider a non-selfadjoint $h$-differential model operator $P_h$ in the semiclassical limit ($h\rightarrow 0$) subject to small random perturbations. Furthermore, we let the coupling constant $\delta$ be $\exp\{-\frac{1}{Ch}\}\leq \delta…
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…
For a class of non-selfadjoint semiclassical pseudodifferential operators with double characteristics, we study bounds for resolvents and estimates for low lying eigenvalues. Specifically, assuming that the quadratic approximations of the…
We consider the fractional Laplacian on a domain and investigate the asymptotic behavior of its eigenvalues. Extending methods from semi-classical analysis we are able to prove a two-term formula for the sum of eigenvalues with the leading…
For a class of PT-symmetric operators with small random perturbations, the eigenvalues obey Weyl asymptotics with probability close to 1. Consequently, when the principal symbol is non-real, there are many non-real eigenvalues.
This is the second in a series of works devoted to small non-selfadjoint perturbations of selfadjoint semiclassical pseudodifferential operators in dimension 2. As in our previous work, we consider the case when the classical flow of the…
We present a survey of some recent results concerning the location and the Weyl formula for the complex eigenvalues of two non self-adjoint operators. We study the eigenvalues of the generator $G$ of the contraction semigroup $e^{tG}, \: t…
We establish eigenfunctions estimates, in the semi-classical regime, for critical energy levels associated to an isolated singularity. For Schr\"odinger operators, the asymptotic repartition of eigenvectors is the same as in the regular…