Rodolfo Figari

The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity

Abstract We consider the Schrodinger equation in R 3 with nonlinearity concentrated in a finite set of points. We formulate the problem in the space of finite energy V, which is strictly larger than the standard H1-space due to the specific singularity exhibited by the solutions. We prove local existence and, for a repulsive or weakly attractive nonlinearity, also global existence of the solutions.

Ionization for Three Dimensional Time-dependent Point Interactions

We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the “strength” of the interaction α(t) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of α(t), we prove that there is complete ionization as t→∞, starting from a bound state at time t=0. Moreover we prove also that, under the same conditions, all the states of the system are scattering states.

Decoherence in a two-particle model

We consider a simple one-dimensional quantum system consisting of a heavy and a light particle interacting via a point interaction. The initial state is chosen to be a product state, with the heavy particle described by a coherent superposition of two spatially separated wave packets with opposite momentum and the light particle localized in the region between the two wave packets. We characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. We derive the corresponding reduced density matrix for the heavy particle and explicitly compute the (partial) decoherence effect for the heavy particle induced by the presence of the light one for a particular set up of the parameters.

On the asymptotic behaviour of a quantum two-body system in the small mass ratio limit

We consider a quantum system of two particles in dimension three interacting via a smooth potential. We characterize the asymptotic dynamics in the limit of small mass ratio for an initial state given in product form, with an explicit control of the error. An application to the decoherence effect produced on the heavy particle is also discussed.

Spin-dependent point potentials in one and three dimensions

We consider a system realized with one spinless quantum particle and an array of N spins 1/2 in dimensions 1 and 3. We characterize all the Hamiltonians obtained as point perturbations of assigned free dynamics in terms of some generalized boundary conditions. For every boundary condition, we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin-dependent point potentials that could be of interest as multi-component solvable models.

Joint excitation probability for two harmonic oscillators in one dimension and the Mott problem

We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions a1 and a2, with a1>0 and ∣a2∣>a1, in the two possible situations: a2>0 and a2 0 at time t>∣a2∣v0−1 when a2 0). We prove that Pn1n2−(t) is negligible with respect to Pn1n2+(t) up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott [“The wave mechanics of α-ray tracks,” Proc. R. Soc. London, Ser. A 126, 79 (1929)], ...

A Time-Dependent Perturbative Analysis for a Quantum Particle in a Cloud Chamber

We consider a simple model of a cloud chamber consisting of a test particle (the α-particle) interacting with two quantum systems (the atoms of the vapor) initially confined around

Resonances in Models of Spin Dependent Point Interactions

{a_1, a_2 \in \mathbb{R}^3}