Related papers: Construction of quasimodes for non-selfadjoint ope…
The pseudospectra (or spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics. In fact, for non-selfadjoint operators the resolvent could be very large outside the spectrum, making the…
We consider nonselfadjoint perturbations of semiclassical harmonic oscillators. Under appropriate dynamical assumptions, we establish some spectral estimates such as upper bounds on the resolvent near the real axis when no geometric control…
In this work, concentration properties of quasimodes for perturbed semiclassical harmonic oscillators are studied. The starting point of this research comes from the fact that, in the presence of resonances between frequencies of the…
We prove a semiclassical resolvent estimate for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime. The proof relies on improved ellipticity properties for the symbol of the…
We estimate the norm of the resolvent of non-selfadjoint Berezin Toeplitz operators in the semi-classical limit, under various assumptions on the Poisson bracket of the real and imaginary parts of the symbol. In case this bracket is…
We build a combinatorial invariant, called the spectral monodromy from the spectrum of a non-selfadjoint h -pseudodifferential operator with two degrees of freedom in the semi-classical limit. We treat small non-selfadjoint perturbation of…
We study semiclassical asymptotics for spectra of non-selfadjoint perturbations of selfadjoint analytic $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable.…
Depending on the behaviour of the complex-valued electromagnetic potential in the neighbourhood of infinity, pseudomodes of one-dimensional Dirac operators corresponding to large pseudoeigenvalues are constructed. This is a first systematic…
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…
For a large class of semiclassical operators $P(h)-z$ which includes Schr\"odinger operators on manifolds with boundary, we construct the Quantum Monodromy operator $M(z)$ associated to a periodic orbit $\gamma$ of the classical flow. Using…
For one-dimensional Schroedinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. Our (non-semi-classical) approach results in substantial progress in achieving optimal conditions…
We compute quasinormal mode frequencies for static limits of physical black holes - semi-classical black hole solutions to Einstein-Hilbert gravity characterized by the finite formation time of an apparent horizon and its weak regularity.…
In this paper we explore a certain class of non-selfadjoint operators acting in a complex separable Hilbert space. We consider a perturbation of a non-selfadjoint operator by an operator that is also non-selfadjoint. Our consideration is…
We consider a class of pseudodifferential operators with a doubly characteristic point, where the quadratic part of the symbol fails to be elliptic but obeys an averaging assumption. Under suitable additional assumptions, semiclassical…
For a class of non-selfadjoint semiclassical operators in dimension one, we get a complete asymptotic description of all eigenvalues near a critical value of the leading symbol of the operator on the boundary of the pseudospectrum.
Here we explore, in a series of articles, semiclassical quasimodes u(h,b), approximative solutions P(h)u(h,b)\sim 0, depending on $0<h<1$, and on b, the subprincipal symbol. We study a pseudodifferential operator with transversal…
We consider a semiclassical (pseudo)differential operator on a compact surface $(M,g)$, such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit $\gamma$ at some energy $E_0$. For any $\epsilon>0$,…
We propose an alternative to the usual time--independent Born--Oppenheimer approximation that is specifically designed to describe molecules with non--symmetrical hydrogen bonds. In our approach, the masses of the hydrogen nuclei are scaled…
Quasinormal modes and frequencies are the eigenvectors and eigenvalues of a non-Hermitian differential operator. They hold crucial significance in the physics of black holes. The analysis of quasinormal modes of black holes in…
The purpose of this note is to review some recent results concerning the pseudospectra and the eigenvalues asymptotics of non-selfadjoint semiclassical pseudo-differential operators subject to small random perturbations.