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 $m<m^* = (13.607)^{-1}$, we give a rigorous proof of the occurrence of the Efimov effect, i.e., the existence of infinitely many negative eigenvalues for the three-particle Hamiltonian $H$. More precisely, we prove that 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 $\mathcal C(m)$.Comment: 26 page

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

From quantum to classical world: emergence of trajectories in a quantum system.

This note deals with models of quantum systems where the emergence of a classical behavior can be concretely analyzed. We first briefly review some well known difficulties arising in the classical limit of quantum mechanics according to the Copenhagen interpretation. Then we discuss the seminal contribution by Mott (1929) on the tracks observed in a cloud chamber, where the problem can be approached in a particularly transparent way. Finally, we propose a model Hamiltonian, with interaction described by spin dependent point interactions, where Mott’s analysis can be rephrased and the result can be rigorously formulated

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 $\alpha$-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

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.Comment: 7 page

System of fermions with zero-range interactions

We discuss the stability problem for a system of N identical fermions with unit mass interacting with a different particle of mass m via zero-range interactions in dimension three. We find a stability parameter m*(N) &gt;0, increasing with N, such that the Hamiltonian of the system is self-adjoint and bounded from below for m&gt;m*(N)

Classical behavior in quantum systems: the case of straight tracks in a cloud chamber

The aim of this review is to discuss in a pedagogical way the problem of the emergence of a classical behavior in certain physical systems which, in principle, are correctly described by quantum mechanics. It is stressed that the limit $\hbar \to 0$ is not sufficient and the crucial role played by the environment must be taken into account. In particular it is recalled the old problem raised by Mott in 1929 (\cite{m}) concerning the straight tracks observed in a cloud chamber, produced by an $\alpha$-particle emitted by a source in the form of a spherical wave. The conceptual relevance of the problem for a clearer understanding of the classical limit is discussed in a historical perspective. Moreover a simple mathematical model is proposed, where the result of Mott is obtained in a rigorous mathematical way.Comment: review paper, 15 page

Joint excitation probability for two harmonic oscillators in dimension one 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 $a_1$, $a_2$, with $a_1 >0$, $|a_2|>a_1$, in the two possible situations: $a_2>0$ and $a_2 <0$. At time zero the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. %$\pm M v_0$. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability $\mathcal{P}^{-}_{n_1 n_2} (t)$ (resp. $\mathcal{P}^{+}_{n_1 n_2} (t)$) that both oscillators are in the excited states labelled by $n_1$, $n_2 >0$ at time $t > |a_2| v_0^{-1}$ when $a_2 0$). We prove that $\mathcal{P}_{n_1 n_2}^- (t)$ is negligible with respect to $\mathcal{P}_{n_1 n_2}^+ (t)$, 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 in \cite{m}, where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schr\"{o}dinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schr\"{o}dinger equation for the three-particle system, with no reference to the wave packet collapse postulate.Comment: 26 page