arXiv subjects

A. Teta

Publications and source records attributed to A. Teta.

10 recordsLinked to original sources

arXiv2026

A Zero-Range Model for the Efimov Effect in the Born-Oppenheimer Approximation

In this note we discuss the Efimov effect emerging in a three-particle quantum system with zero-range interactions. In particular, we consider two non-interacting identical bosons plus a different lighter particle such that the interaction between a boson and the light particle is resonant. We also assume the validity of the Born-Oppenheimer approximation. Under these conditions, we show that the three-particle system exhibits infinitely many negative eigenvalues which accumulate at zero and satisfy the universal geometrical law characterising the Efimov effect. The result we find is a generalisation of previous results recently obtained in [13, 24].

math-ph↗

arXiv2023

A new look at the theory of point interactions

We investigate the entire family of multi-center point interaction Hamiltonians. We show that a large sub-family of these operators do not become either singular or trivial when the positions of two or more scattering centers tend to coincide. In this sense, they appear to be renormalised by default as opposed to the "local" point interaction Hamiltonians usually considered in the literature as the ones of physical interest. In the two-center case we study the behaviour of the negative eigenvalues as a function of the center distance. The result is used to analyze a formal Born-Oppenheimer approximation of a three-particle system with two heavy and one light particle. We show that this simplified model does not show any ultra-violet catastrophe and we prove that the ratio of successive low energy eigenvalues follows the Efimov geometrical law.

math-ph↗

arXiv2019

A Quantum Model of Feshbach Resonances

We consider a quantum model of two-channel scattering to describe the mechanism of a Feshbach resonance. We perform a rigorous analysis in order to count and localize the energy resonances in the perturbative regime, i.e., for small inter-channel coupling, and in the non-perturbative one. We provide an expansion of the effective scattering length near the resonances, via a detailed study of an effective Lippmann-Schwinger equation with energy-dependent potential.

math-ph↗

arXiv2015

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^* \simeq (13.607)^{-1} $ a self-adjoint and lower bounded Hamiltonian $ H_0 $ 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 \in( m^*,m^{**}) $, where $ m^{**} \simeq (8.62)^{-1} $, there is a further family of self-adjoint and lower bounded Hamiltonians $ H_{0,β} $, $ β\in \mathbb{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.

math-ph↗

arXiv2014

The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit

In the present paper we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: \[ i\frac{d}{dt} ψ^{\varepsilon}(t) =-Δψ^{\varepsilon}(t) + \frac{1}εV\left(\frac{x}ε\right)|ψ^{\varepsilon}(t)|^{2μ}ψ^{\varepsilon}(t) \quad \quad ε>0\ ,\quad V\in L^1(\mathbb{R},(1+|x|)dx) \cap L^\infty(\mathbb{R}) \ . \] This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit $ε\to 0$, the weak (integral) dynamics converges in $H^1(\mathbb{R})$ to the weak dynamics of the NLS with point-concentrated nonlinearity: \[ i\frac{d}{dt} ψ(t) =H_αψ(t) . \] where $H_α$ is the laplacian with the nonlinear boundary condition at the origin $ψ'(t,0+)-ψ'(t,0-)=α|ψ(t,0)|^{2μ}ψ(t,0)$ and $α=\int_{\mathbb{R}}Vdx$. The convergence occurs for every $μ\in \mathbb{R}^+$ if $V \geq 0$ and for every $μ\in (0,1)$ otherwise. The same result holds true for a nonlinearity with an arbitrary number $N$ of concentration points

math-ph↗

arXiv2012

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 \geq 2 $ identical fermions, with unit mass, interacting with a different particle, with mass $ m $, via a zero-range interaction of strength $ α\in \R $. We construct the corresponding renormalised quadratic (or energy) form $ \form $ and the so-called Skornyakov-Ter-Martirosyan symmetric extension $ H_α $, 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 $ \form $ is closed and bounded from below. As a consequence, $ \form $ defines a unique self-adjoint and bounded from below extension of $ H_α$ and therefore the system is stable. On the other hand, we also show that the form $ \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.

math-ph↗

arXiv2009

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 alpha-particle) interacting with two other particles (the atoms of the vapour) subject to attractive potentials centered in $a_1, a_2 \in \mathbb{R}^3$. At time zero the alpha-particle is described by an outgoing spherical wave centered in the origin and the atoms are in their ground state. We show that, under suitable assumptions on the physical parameters of the system and up to second order in perturbation theory, the probability that both atoms are ionized is negligible unless $a_2$ lies on the line joining the origin with $a_1$. The work is a fully time-dependent version of the original analysis proposed by Mott in 1929.

math-ph↗

arXiv2002

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.

quant-ph↗

arXiv1999

The Flux-Across-Surfaces Theorem for a Point Interaction Hamiltonian

The flux-across-surfaces theorem establishes a fundamental relation in quantum scattering theory between the asymptotic outgoing state and a quantity which is directly measured in experiments. We prove it for a hamiltonian with a point interaction, using the explicit expression for the propagator. The proof requires only assuptions on the initial state and it covers also the case of zero-energy resonance. We also outline a different approach based on generalized eigenfunctions, in view of a possible extension of the result.

math-ph↗

arXiv1999

Schrodinger Equation with Moving Point Interactions in Three Dimensions

We consider the motion of a non relativistic quantum particle in R^3 subject to n point interactions which are moving on given smooth trajectories. Due to the singular character of the time-dependent interaction, the corresponding Schrodinger equation does not have solutions in a strong sense and, moreover, standard perturbation techniques cannot be used. Here we prove that, for smooth initial data, there is a unique weak solution by reducing the problem to the solution of a Volterra integral equation involving only the time variable. It is also shown that the evolution operator uniquely extends to a unitary operator in $L^{2}(R^{3})$.

math-ph↗