Analytic linearization of nonlinear perturbations of Fuchsian systems

Nonlinear perturbation of Fuchsian systems are studied in regions including two singularities. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable). Nevertheless, it is shown that in the case when the linear part has commuting monodromy, and the eigenvalues have positive real parts, there exists a unique correction function of the nonlinear part so that the corrected system becomes analytically linearizable

Evolution of a model quantum system under time periodic forcing: conditions for complete ionization

We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation $\eta(t)$. We show that for generic $\eta(t)$, which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as $t\to\infty$. This is irrespective of the magnitude or frequency (resonant or not) of $\eta(t)$. There are however exceptional, very non-generic $\eta(t)$, that do not lead to full ionization, which include rather simple explicit periodic functions. For these $\eta(t)$ the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator

Decay of a Bound State under a Time-Periodic Perturbation: a Toy Case

We study the time evolution of a three dimensional quantum particle, initially in a bound state, under the action of a time-periodic zero range interaction with ``strength'' (\alpha(t)). Under very weak generic conditions on the Fourier coefficients of (\alpha(t)), we prove complete ionization as (t \to \infty). We prove also that, under the same conditions, all the states of the system are scattering states.Comment: LaTeX2e, 15 page

Nonlinear Schroedinger equation with two symmetric point interactions in one dimension

We consider a time-dependent one-dimensional nonlinear Schroedinger equation with a symmetric potential double well represented by two delta interactions. Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

Ionization in damped time-harmonic fields

We study the asymptotic behavior of the wave function in a simple one dimensional model of ionization by pulses, in which the time-dependent potential is of the form $V(x,t)=-2\delta(x)(1-e^{-\lambda t} \cos\omega t)$, where $\delta$ is the Dirac distribution. We find the ionization probability in the limit $t\to\infty$ for all $\lambda$ and $\omega$. The long pulse limit is very singular, and, for $\omega=0$, the survival probability is $const \lambda^{1/3}$, much larger than $O(\lambda)$, the one in the abrupt transition counterpart, $V(x,t)=\delta(x)\mathbf{1}_{\{t\ge 1/\lambda\}}$ where $\mathbf{1}$ is the Heaviside function