Search results for "35B34"
showing 4 items of 4 documents
Resonances over a potential well in an island
2020
In this paper we study the distribution of scattering resonances for a multidimensional semi-classical Schr\"odinger operator, associated to a potential well in an island at energies close to the maximal one that limits the separation of the well and the surrounding sea.
Adiabatic evolution and shape resonances
2017
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…
Resonances for nonanalytic potentials
2009
We consider semiclassical Schr"odinger operators on $R^n$, with $C^infty$ potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a non-analytic framework. Here, under some additional conditions, we show that resonances are invariantly defined up to any power of their imaginary part. The theory is based on resolvent estimates for families of approximating distorted operators with potentials that are holomorphic in narrow complex sectors around $R^n$.
Fractal Weyl law for open quantum chaotic maps
2014
We study the semiclassical quantization of Poincar\'e maps arising in scattering problems with fractal hyperbolic trapped sets. The main application is the proof of a fractal Weyl upper bound for the number of resonances/scattering poles in small domains near the real axis. This result encompasses the case of several convex (hard) obstacles satisfying a no-eclipse condition.