Martin Vogel - Random non-selfadjoint operator, tunneling and exponentially small singular values of the d-bar operator

lundi 23 mars 2026, 13:00 → 15:15 Europe/Paris

435 (UMPA, ENS de Lyon)

The spectra of non-selfadjoint operators, unlike their selfadjoint counterparts, can be 
highly sensitive to even small perturbations. This spectral instability, although an adversary 
for numerical methods, can lead to exciting spectral phenomena in the presence of 
small random perturbations. 

In the first part of the talk I will review some recent developments in the spectral theory 
of non-selfadjoint operators subject to small random perturbations. In particular we are 
interested in studying the behavior of the eigenvectors associated with the discrete spectrum. 
A major ingredient in this analysis is the behavior of the singular vectors of the unperturbed 
operator associated with its small singular spectrum. 

In the second part I will present a resent result studying  the exponentially small singular 
values (and the associated singular vectors) of the semiclassical d-bar operator on an 
exponentially weighted L^{^2} space on a compact Riemann surface. We will assume that 
the Laplacian of the exponential weight changes sign along a curve.
We will introduce the notion of upper and lower bound weights which
give together with the orthogonal Bergman projection precise upper and lower
bounds on the number of small singular values. Solving a free boundary value
problem we obtain optimal weights which yield Weyl asymptotics for the
counting function of exponentially small singular values. We also provide
a precise description of the leading term of the Weyl asymptotics in the
regime of small exponential decay rates of the singular values.

This talk is based on joint work with J. Söstrand and M. Hitrik.