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Alessandro Michelangeli

Publications and source records attributed to Alessandro Michelangeli.

46 records · Page 3Linked to original sources
arXiv2018

Spectral analysis of the 2+1 fermionic trimer with contact interactions

We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise the discrete spectrum and prove its finiteness, qualify the angular symmetry of the eigenfunctions, and prove the increasing monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence or absence of bound states in the physically relevant regimes of masses.

math-ph↗
arXiv2017

Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan-- Skornyakov type

We reconstruct the whole family of self-adjoint Hamiltonians of Ter-Martirosyan-- Skornyakov type for a system of two identical fermions coupled with a third particle of different nature through an interaction of zero range. We proceed through an operator-theoretic approach based on the self-adjoint extension theory of Krein, Visik, and Birman. We identify the explicit Krein-Visik-Birman extension parameter as an operator on the "space of charges" for this model (the "Krein space") and we come to formulate a sharp conjecture on the dimensionality of its kernel. Based on our conjecture, for which we also discuss an amount of evidence, we explain the emergence of a multiplicity of extensions in a suitable regime of masses and we reproduce for the first time the previous partial constructions obtained by means of an alternative quadratic form approach.

math-ph↗
arXiv2017

Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials

We prove the existence of weak solutions in the space of energy for a class of non-linear Schroedinger equations in the presence of a external rough magnetic potential. Under our assumptions it is not possible to study the problem by means of usual arguments like resolvent techniques or Fourier integral operators, for example. We use a parabolic regularization and we solve the approximating Cauchy problem. This is achieved by obtaining suitable smoothing estimates for the dissipative evolution. The total mass and energy bounds allow to extend the solution globally in time. We then infer sufficient compactness properties in order to produce a global-in-time finite energy weak solution to our original problem.

math.AP↗
arXiv2017

Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates

We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.

math-ph↗
arXiv2017

On fractional powers of singular perturbations of the Laplacian

We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.

math.FA↗
arXiv2017

On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians

For quantum systems of zero-range interaction we discuss the mathematical scheme within which modelling the two-body interaction by means of the physically relevant ultra-violet asymptotics known as the "Ter-Martirosyan-Skornyakov condition" gives rise to a self-adjoint realisation of the corresponding Hamiltonian. This is done within the self-adjoint extension scheme of Krein, Visik, and Birman. We show that the Ter-Martirosyan-Skornyakov asymptotics is a condition of self-adjointness only when is imposed in suitable functional spaces, and not just as a pointwise asymptotics, and we discuss the consequences of this fact on a model of two identical fermions and a third particle of different nature.

math-ph↗
arXiv2016

Mean-field quantum dynamics for a mixture of Bose-Einstein condensates

We study the effective time evolution of a large quantum system consisting of a mixture of different species of identical bosons in interaction. If the system is initially prepared so as to exhibit condensation in each component, we prove that condensation persists at later times and we show quantitatively that the many-body Schrödinger dynamics is effectively described by a system of coupled cubic non-linear Schrödinger equations, one for each component.

math-ph↗
arXiv2011

Dynamical Collapse of Boson Stars

We study the time evolution in system of $N$ bosons with a relativistic dispersion law interacting through an attractive Coulomb potential with coupling constant $G$. We consider the mean field scaling where $N$ tends to infinity, $G$ tends to zero and $λ= G N$ remains fixed. We investigate the relation between the many body quantum dynamics governed by the Schrödinger equation and the effective evolution described by a (semi-relativistic) Hartree equation. In particular, we are interested in the super-critical regime of large $λ$ (the sub-critical case has been studied in \cite{ES,KP}), where the nonlinear Hartree equation is known to have solutions which blow up in finite time. To inspect this regime, we need to regularize the Coulomb interaction in the many body Hamiltonian with an $N$ dependent cutoff that vanishes in the limit $N\to \infty$. We show, first, that if the solution of the nonlinear equation does not blow up in the time interval $[-T,T]$, then the many body Schrödinger dynamics (on the level of the reduced density matrices) can be approximated by the nonlinear Hartree dynamics, just as in the sub-critical regime. Moreover, we prove that if the solution of the nonlinear Hartree equation blows up at time $T$ (in the sense that the $H^{1/2}$ norm of the solution diverges as time approaches $T$), then also the solution of the linear Schrödinger equation collapses (in the sense that the kinetic energy per particle diverges) if $t \to T$ and, simultaneously, $N \to \infty$ sufficiently fast. This gives the first dynamical description of the phenomenon of gravitational collapse as observed directly on the many body level.

math-ph↗
arXiv2008

Dynamical formation of correlations in a Bose-Einstein condensate

We consider the evolution of $N$ bosons interacting with a repulsive short range pair potential in three dimensions. The potential is scaled according to the Gross-Pitaevskii scaling, i.e. it is given by $N^2V(N(x_i-x_j))$. We monitor the behavior of the solution to the $N$-particle Schrödinger equation in a spatial window where two particles are close to each other. We prove that within this window a short scale interparticle structure emerges dynamically. The local correlation between the particles is given by the two-body zero energy scattering mode. This is the characteristic structure that was expected to form within a very short initial time layer and to persist for all later times, on the basis of the validity of the Gross-Pitaevskii equation for the evolution of the Bose-Einstein condensate. The zero energy scattering mode emerges after an initial time layer where all higher energy modes disperse out of the spatial window. We can prove the persistence of this structure up to sufficiently small times before three-particle correlations could develop.

math-ph↗