arXiv subjects

Daniele Ferretti

Publications and source records attributed to Daniele Ferretti.

6 recordsLinked to original sources

arXiv2025

Effective Dynamics for Weakly Interacting Bosons in an Iterated High-Density Thermodynamic Limit

We study the time evolution of weakly interacting Bose gases on a three-dimensional torus of arbitrary volume. The coupling constant is supposed to be inversely proportional to the density, which is considered to be large and independent of the particle number. We take into account a class of initial states exhibiting quasi-complete Bose-Einstein condensation. For each fixed time in a finite interval, we prove the convergence of the one-particle reduced density matrix towards the projection onto the normalised order parameter describing the condensate - evolving according to the Hartree equation - in the iterated limit where the volume (and therefore the particle number), and subsequently the density go to infinity. The rate of convergence depends only on the density and on the decay of both the expected number of particles and the energy of the initial quasi-vacuum state.

math-ph↗

arXiv2025

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, Høegh-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↗

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 $δ$) 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↗

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 $\;E_{n+1} / E_n \; \to \; e^{-\frac{2π}{s_0}}\,\; \,\text{for} \,\; n\to +\infty$ holds, where $s_0\approx 1.00624$.

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 $γ$ of the three-body force is larger than a threshold parameter $γ_c$. Moreover, we give an alternative and much simpler proof of the above result whenever $γ> γ'_c$, with $γ'_c$ strictly larger than $γ_c$. Finally, we show that the threshold value $γ_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $γ<γ_c$.

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.

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