arXiv subjects

Alessandro Teta

Publications and source records attributed to Alessandro Teta.

At least 19 recordsLinked to original sources

arXiv2026

On Fermi's model for the scattering of a slow neutron from a bound proton

We consider a model Hamiltonian, introduced by Fermi in 1936, describing a two-particle system made of a neutron and a harmonically bound proton, where the neutron-proton interaction has the form of a $\delta$-potential. For such Hamiltonian we prove the Limiting Absorption Principle and describe the stationary scattering theory. Finally, we derive Fermi's formula for the scattering cross-section valid in the Born approximation.

math-ph↗

arXiv2026

A mathematical model for the Einstein-Podolsky-Rosen argument

We study a nonrelativistic system made of two quantum particles constrained to move on a line and a spin located at a fixed point of the line. Initially the two particles are in a maximally entangled state and the spin is down. The first particle interacts with the spin while the second particle is free, i.e., it does not interact neither with the first particle nor with the spin. We rigorously prove that there is a correlation between the state of the spin and the state of the second particle. More precisely, we show that, in a suitable scaling limit, if the first particle flips the spin, then the second particle possesses a definite momentum in the direction opposite to the spin.

math-ph↗

arXiv2024

Hamiltonians for Quantum Systems with Contact Interactions

We discuss the problem of constructing self-adjoint and lower bounded Hamiltonians for a system of $n>2$ non-relativistic quantum particles in dimension three with contact (or zero-range or $\delta$) interactions. Such interactions are described by (singular) boundary conditions satisfied at the coincidence hyperplanes, i.e., when the coordinates of two particles coincide. Following the line of recent works appeared in the literature, we introduce a boundary condition slightly modified with respect to usual boundary condition one has in the one-body problem. With such new boundary condition we can show that the instability property due to the fall to the center phenomenon described by Minlos and Faddeev in 1962 is avoided. Then one obtains a physically reasonable Hamiltonian for the system. We apply the method to the case of a gas of $N$ interacting bosons and to the case of $N$ distinguishable particles of equal mass $M$ interacting with a different particle. In the latter case we also discuss the limit of the model for $M \longrightarrow +\infty$. We show that in the limit one obtains the one-body Hamiltonian for the light particle subject to $N$ (non-local) point interactions placed at fixed positions. We will verify that such non-local point interactions do not exhibit the ultraviolet pathologies that are present in the case of standard local point interactions.

math-ph↗

arXiv2024

Hamiltonian for a Bose gas with Contact Interactions

We study the Hamiltonian for a three-dimensional Bose gas of $N \geq 3$ spinless particles interacting via zero-range (also known as contact) interactions. Such interactions are encoded by (singular) boundary conditions imposed on the coincidence hyperplanes, i.e., when the coordinates of two particles coincide. It is well known that imposing the same kind of boundary conditions as in the two-body problem with a point interaction leads to a Hamiltonian unbounded from below (and thus unstable). This is due to the fact that the interaction becomes overly strong and attractive when the coordinates of three or more particles coincide. In order to avoid such instability, we develop a suggestion originally formulated by Minlos and Faddeev in 1962, introducing slightly modified boundary conditions that weaken the strength of the interaction between two particles $i$ and $j$ in two scenarios: (a) a third particle approaches the common position of $i$ and $j$; (b) another distinct pair of particles approach each other. In all other cases, the usual boundary condition is restored. Using a quadratic form approach, we construct a class of Hamiltonians characterized by such modified boundary conditions, that are self-adjoint and bounded from below. We also compare our approach with the one developed years ago by Albeverio, H{\o}egh-Krohn and Streit using the theory of Dirichlet forms (J. Math. Phys., 18, 907--917, 1977). In particular, we show that the $N$-body Hamiltonian defined by Albeverio et al. is a special case of our class of Hamiltonians. Furthermore, we also introduce a Dirichlet form by considering a more general weight function, and we prove that the corresponding $N$-body Hamiltonians essentially coincide with those constructed via our method.

math-ph↗

arXiv2023

Rigorous derivation of the Efimov effect in a simple model

We consider a system of three identical bosons in $\mathbb{R}^3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$. Using a quadratic form approach we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of $a$. In particular this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, \emph{i.e.}, we show that the Hamiltonian has an infinite sequence of negative eigenvalues $E_n$ accumulating at zero and fulfilling the asymptotic geometrical law $\;E_{n+1} / E_n \; \to \; e^{-\frac{2\pi}{s_0}}\,\; \,\text{for} \,\; n\to +\infty$ holds, where $s_0\approx 1.00624$.

math-ph↗

arXiv2022

Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity

We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on $\mathbb{S}^2$. Precisely, local well-posedness is proved for any $C^2$ power-nonlinearity, while global well-posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point-concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas.

math.AP↗

arXiv2022

Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three

We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan--Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non local interactions, also called separable potentials, with a suitably renormalized coupling constant.

math-ph↗

arXiv2022

Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions

We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In order to avoid the well known instability phenomenon, we consider the so-called Minlos-Faddeev regularization of such Hamiltonian, heuristically corresponding to the introduction of a three-body repulsion. We review the main concerning results recently obtained. In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $\mathcal H$ can be constructed provided that the strength $\gamma$ of the three-body force is larger than a threshold parameter $\gamma_c$. Moreover, we give an alternative and much simpler proof of the above result whenever $\gamma > \gamma'_c$, with $\gamma'_c$ strictly larger than $\gamma_c$. Finally, we show that the threshold value $\gamma_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $\gamma<\gamma_c$.

math-ph↗

arXiv2022

Regularized Zero-Range Hamiltonian for a Bose Gas with an Impurity

We study the Hamiltonian for a system of N identical bosons interacting with an impurity, i.e., a different particle, via zero-range forces in dimension three. It is well known that, following the standard approach, one obtains the Ter-Martirosyan Skornyakov Hamiltonian which is unbounded from below. In order to avoid such instability problem, we introduce a three-body force acting at short distances. The effect of this force is to reduce to zero the strength of the zero-range interaction between two particles, i.e., the impurity and a boson, when another boson approaches the common position of the first two particles. We show that the Hamiltonian defined with such regularized interaction is self-adjoint and bounded from below if the strength of the three-body force is sufficiently large. The method of the proof is based on a careful analysis of the corresponding quadratic form.

math-ph↗

arXiv2020

On the Hamiltonian for three bosons with point interactions

We briefly summarize the most relevant steps in the search of rigorous results about the properties of quantum systems made of three bosons interacting with zero-range forces. We also describe recent attempts to solve the unboundedness problem of point-interaction Hamiltonians for a three-boson system, keeping unaltered the spectrum structure at low energies.

math-ph↗

arXiv2018

The three-body problem in dimension one: From short-range to contact interactions

We consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in norm resolvent sense. The two-body rescaled potentials are of the form $v^{\varepsilon}_σ(x_σ)= \varepsilon^{-1} v_σ(\varepsilon^{-1}x_σ)$, where $σ= 23, 12, 31$ is an index that runs over all the possible pairings of the three particles, $x_σ$ is the relative coordinate between two particles, and $\varepsilon$ is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials $v_σ$ with $α_σδ_σ$, where $δ_σ$ is the Dirac delta-distribution centered on the coincidence hyperplane $x_σ=0$ and $α_σ= \int_{\mathbb{R}} v_σdx_σ$. To prove the convergence of the resolvents we make use of Faddeev's equations.

math-ph↗

arXiv2017

Universal low-energy behavior in a quantum Lorentz gas with Gross-Pitaevskii potentials

We consider a quantum particle interacting with $N$ obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form $N^2 V(Nx)$ (Gross-Pitaevskii potential). We show convergence of the $N$ dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for $N$ large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.

math-ph↗

arXiv2017

The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

We consider a scaling limit of a nonlinear Schrödinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equation \begin{equation*} i\frac{\partial }{\partial t} ψ^\varepsilon(t)= -Δψ^\varepsilon(t) + g(\varepsilon,μ,|(ρ^\varepsilon,ψ^\varepsilon(t))|^{2μ}) (ρ^\varepsilon,ψ^\varepsilon(t)) ρ^\varepsilon \end{equation*} where $ρ^{\varepsilon} \to δ_0$ weakly and the function $g$ embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics. The limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well-posedness, blow-up and asymptotic properties of solutions. Our result is the first justification of the model as the point limit of a regularized dynamics.

math-ph↗

arXiv2017

Efimov effect for a three-particle system with two identical fermions

We consider a three-particle quantum system in dimension three composed of two identical fermions of mass one and a different particle of mass $m$. The particles interact via two-body short range potentials. We assume that the Hamiltonians of all the two-particle subsystems do not have bound states with negative energy and, moreover, that the Hamiltonians of the two subsystems made of a fermion and the different particle have a zero-energy resonance. Under these conditions and for $m m^*$ the number of negative eigenvalues of $H$ is finite and for $m<m^*$ the number $N(z)$ of negative eigenvalues of $H$ below $z<0$ has the asymptotic behavior $N(z) \sim \mathcal C(m) |\log|z||$ for $z \rightarrow 0^-$. Moreover, we give an upper and a lower bound for the positive constant $\mathcal C(m)$.

math-ph↗

arXiv2016

On the quantum mechanical three-body problem with zero-range interactions

In this note we discuss the quantum mechanical three-body problem with pairwise zero-range interactions in dimension three. We review the state of the art concerning the construction of the corresponding Hamiltonian as a self-adjoint operator in the bosonic and in the fermionic case. Exploiting a quadratic form method, we also prove self-adjointness and boundedness from below in the case of three identical bosons when the Hilbert space is suitably restricted, i.e., excluding the "s-wave" subspace.

math-ph↗

arXiv2015

Energy lower bound for the unitary N+1 fermionic model

We consider the stability problem for a unitary N+1 fermionic model, i.e., a system of $N$ identical fermions interacting via zero-range interactions with a different particle, in the case of infinite two-body scattering length. We present a slightly more direct and simplified proof of a recent result obtained in \cite{CDFMT}, where a sufficient stability condition is proved under a suitable assumption on the mass ratio.

cond-mat.quant-gas↗

arXiv2013

Semiclassical wave-packets emerging from interaction with an environment

We study the quantum evolution in dimension three of a system composed by a test particle interacting with an environment made of $N$ harmonic oscillators. At time zero the test particle is described by a spherical wave, i.e. a highly correlated continuous superposition of states with well localized position and momentum, and the oscillators are in the ground state. Under suitable assumptions on the physical parameters characterizing the model, we give an asymptotic expression of the solution of the Schrödinger equation of the system with an explicit control of the error. The result shows that the approximate expression of the wave function is the sum of two terms, orthogonal in $L^2(\erre^{3(N+1)})$ and describing rather different situations. In the first one all the oscillators remain in their ground state and the test particle is described by the free evolution of a slightly deformed spherical wave. The second one consists of a sum of $N$ terms where in each term there is only one excited oscillator and the test particle is correspondingly described by the free evolution of a wave packet, well concentrated in position and momentum. Moreover the wave packet emerges from the excited oscillator with an average momentum parallel to the line joining the oscillator with the center of the initial spherical wave. Such wave packet represents a semiclassical state for the test particle, propagating along the corresponding classical trajectory. The main result of our analysis is to show how such a semiclassical state can be produced, starting from the original spherical wave, as a result of the interaction with the environment.

math-ph↗

arXiv2012

Emergence of classical trajectories in quantum systems: the cloud chamber problem in the analysis of Mott (1929)

We analyze the paper "The wave mechanics of $α$-ray tracks" (Mott, 1929), published in 1929 by N.F. Mott. In particular, we discuss the theoretical context in which the paper appeared and give a detailed account of the approach used by the author and the main result attained. Moreover, we comment on the relevance of the work not only as far as foundations of Quantum Mechanics are concerned but also as the earliest pioneering contribution in decoherence theory.

math-ph↗