Alessandro Teta

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.

Singular Schrödinger Operators as Limits Point Interaction Hamiltonians

In this paper we give results on the approximation of (generalized) Schrödinger operators of the form - ▵ + µ for some finite Radon measure µ on Rd. For d = 1 we shall show that weak convergence of measures µn to µ implies norm resolvent convergence of the operators -▵ + µn to -▵ + µ. In particular Schrödinger operators of the form - ▵ + µ for some finite Radon measure µ can be regularized or approximated by Hamiltonians describing point interactions. For d = 3 we shall show that a fairly large class of singular interactions can be regarded as limit of point interactions.

A Class of Hamiltonians for a Three-Particle Fermionic System at Unitarity

We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass m, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for m larger than a critical value m∗ ≃ (13.607)−1 a self-adjoint and lower bounded Hamiltonian H0 can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for m ∈ (m∗,m∗∗), where m∗∗ ≃ (8.62)−1, there is a further family of self-adjoint and lower bounded Hamiltonians H0,β, β ∈ ℝ, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.

Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions

We study the stability problem for a non-relativistic quantum system in dimension three composed by N ≥ 2 identical fermions, with unit mass, interacting with a different particle, with mass m, via a zero-range interaction of strength α ∈ ℝ. We construct the corresponding renormalized quadratic (or energy) form and the so-called Skornyakov–Ter–Martirosyan symmetric extension Hα, which is the natural candidate as Hamiltonian of the system. We find a value of the mass m*(N) such that for m > m*(N) the form is closed and bounded from below. As a consequence, defines a unique self-adjoint and bounded from below extension of Hα and therefore the system is stable. On the other hand, we also show that the form is unbounded from below for m < m*(2). In analogy with the well-known bosonic case, this suggests that the system is unstable for m < m*(2) and the so-called Thomas effect occurs.

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.

Quadratic Forms for the Fermionic Unitary Gas Model

We consider a quantum system in dimension three composed by a group of N identical fermions, with mass 1/2, interacting via zero-range interaction with a group of M identical fermions of a different type, with mass m/2. Exploiting a renormalization procedure, we construct the corresponding quadratic form and define the so-called Skornyakov-Ter-Martirosyan extension Hα, which is the natural candidate as a possible Hamiltonian of the system. It is shown that if the form is unbounded from below then Hα is not a self-adjoint and bounded from below operator, and this in particular suggests that the so-called Thomas effect could occur. In the special case N = 2, M = 1 we prove that this is in fact the case when a suitable condition on the parameter m is satisfied.

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