Eigenvalues of non self-adjoint Toeplitz operators

par Nathan Réguer (IRMAR - Université de Rennes)

lundi 4 mai 2026, 14:00 → 15:00 Europe/Paris

Salle Pellos

We are interested in semiclassical operators built from quantisation of symbols $Op_{\hbar}(f)$, in particular on their eigenvalues. Actually, eigenvalues are the poles of the resolvent, and for self-adjoint operators, the norm of the resolvent at a number $E$ is equal to the inverse of the distance from $E$ to the spectrum. Hence, finding quasimodes, in other words eigenfunctions up to a small remainder with respect to $\hbar$: $Op_{\hbar}(f)u_{\hbar} = E u_{\hbar} + O(\hbar^{\infty})$, gives eigenvalues up to a small remainder too. Although, it is no longer true for non self-adjoint operators. Fortunately, it is still possible to get information from exponentially close quasimodes $Op_{\hbar}(f)u_{\hbar} = E u_{\hbar} + O(e^{-\frac{c}{\hbar}})$.
First I will present my problem, which is to find the shape of the spectrum near a point $f(x_0)$ such that $df(x_0)=0$ and the Hessian of $f$ at $x_0$ satisfies an ellipticity condition. Then I will present the framework I use for this study: Toeplitz operators, which is convenient as the state functions are defined directly on the phase space. For these operators, we will see the specific case of quadratic symbols, for which the spectrum is well-known since a long time, but the main ingredient for our method will appear in the proof. To treat the general symbol, we will consider Fourier integral operators with complex phase, and symbolic calculus on Toeplitz operators, with the specificity that all the data have to be analytic. In a last part, I will give some elements of the proof of the main result.