0000000000730589
AUTHOR
Johannes Sjoestrand
Adiabatic evolution and shape resonances
Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we consider a general linear adiabatic evolution problem for a semi-classical Schrödinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter ε \varepsilon with ln ε ≍ − 1 / h \ln \varepsilon \asymp -1/h , where h h denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length ε − N \varepsilon ^{-N} with an error O ( ε N ) {\mathcal O}(\varepsilon ^N) . Here N > 0 N>0 is arbitrary. \center Résumé \endcenter Motivés par un pro…
Weyl law for semi-classical resonances with randomly perturbed potentials
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 certain classes of small random perturbations and show that with probability very close to 1, the number of resonances in rectangles $[a,b]-i[0,ch^{2/3}[$, is equal to the number of eigenvalues in $[a,b]$ of the Dirichlet realization of the unperturbed operator in ${\cal O}$ up to a small remainder.
Interior eigenvalue density of large bi-diagonal matrices subject to random perturbations
We study the spectrum of large a bi-diagonal Toeplitz matrix subject to a Gaussian random perturbation with a small coupling constant. We obtain a precise asymptotic description of the average density of eigenvalues in the interior of the convex hull of the range symbol.
Characterizing boundedness of metaplectic Toeplitz operators
We study Toeplitz operators on the Bargmann space, with Toeplitz symbols given by exponentials of complex quadratic forms. We show that the boundedness of the corresponding Weyl symbols is necessary for the boundedness of the operators, thereby completing the proof of the Berger-Coburn conjecture in this case. We also show that the compactness of such Toeplitz operators is equivalent to the vanishing of their Weyl symbols at infinity.
Two Minicourses on Analytic Microlocal Analysis
These notes correspond roughly to the two minicourses prepared by the authors for the workshop on Analytic Microlocal Analysis, held at Northwestern University in May 2013. The first part of the text gives an elementary introduction to some global aspects of the theory of metaplectic FBI transforms, while the second part develops the general techniques of the analytic microlocal analysis in exponentially weighted spaces of holomorphic functions.