DSpace Repository :: Browsing by Author "Michelangeli, Alessandro"

Browsing by Author "Michelangeli, Alessandro"

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Born approximation in the problem of the rigorous derivation of the Gross-Pitaevskii equation
(2006-04-18T13:24:52Z) Michelangeli, Alessandro; Mathematics; Mathematical Physics
"It has a flavour of Mathematical Physics..."With these words, just few years ago, prof. Di Giacomo used to introduce the topic of the Born approximation within a nonrelativistic potential theory, in his `oversize' course of Theoretical Physics in Pisa. Something maybe too fictitious inside the formal theory of the scattering he was teaching us at that point of the course. Now that I'm (studying to become) a Mathematical Physicist indeed, dealing with such an `exotic tasting' topic, those words come back to the mind, into a new perspective. Here the very recent problem of the rigorous derivation of the cubic nonlinear Schrödinger equation (the Gross-Pitaevskiî equation) is reviewed and discussed, with respect to the role of the Born approximation that one ends up with in an appropriate scaling limit
Bose-Einstein condensation: analysis of problems and rigorous results
(2007-10-08T06:59:18Z) Michelangeli, Alessandro; Mathematics; Mathematical Physics
A class of Hamiltonians for a three-particle fermionic system at unitarity
(2015-05-15) Correggi, Michele; Dell'Antonio, Gianfausto; Finco, Domenico; Michelangeli, Alessandro; Teta, Alessandro; Mathematics
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 $H_{0,\beta}$, $\beta \in \mathbb{R}$, 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.
Convergence of the conjugate gradient method with unbounded operators
(2019-08-27) Caruso, Noe; Michelangeli, Alessandro
In the framework of inverse linear problems on infinite-dimensional Hilbert space, we prove the convergence of the conjugate gradient iterates to an exact solution to the inverse problem in the most general case where the self-adjoint, non-negative operator is unbounded and with minimal, technically unavoidable assumptions on the initial guess of the iterative algorithm. The convergence is proved to always hold in the Hilbert space norm (error convergence), as well as at other levels of regularity (energy norm, residual, etc.) depending on the regularity of the iterates. We also discuss, both analytically and through a selection of numerical tests, the main features and differences of our Convergence result as compared to the case, already available in the literature, where the operator is bounded.
Discrete spectra for critical Dirac-Coulomb Hamiltonians
(2017) Gallone, Matteo; Michelangeli, Alessandro; Mathematics
The one-particle Dirac Hamiltonian with Coulomb interaction is known to be realised, in a regime of large (critical) couplings, by an infinite multiplicity of distinct self-adjoint operators, including a distinguished physically most natural one. For the latter, Sommerfeld’s celebrated fine structure formula provides the well-known expression for the eigenvalues in the gap of the continuum spectrum. Exploiting our recent general classification of all other self-adjoint realisations, we generalise Sommerfeld’s formula so as to determine the discrete spectrum of all other self-adjoint versions of the Dirac-Coulomb Hamiltonian. Such discrete spectra display naturally a fibred structure, whose bundle covers the whole gap of the continuum spectrum.
Dynamics on a graph as the limit of the dynamics on a "fat graph"
(SISSA, 2014) Dell'Antonio, Gianfausto; Michelangeli, Alessandro
We discuss how the vertex boundary conditions for the dynamics of a quantum particle constrained on a graph emerge in the limit of the dynamics of a particle in a tubular region around the graph (\fat graph") when the transversal section of this region shrinks to zero. We give evidence of the fact that if the limit dynamics exists and is induced by the Laplacian on the graph with certain self-adjoint boundary conditions, such conditions are determined by the possible presence of a zero energy resonance on the fat graph. Pictorially, one may say that in the shrinking limit the resonance acts as a bridge connecting the boundary values at the vertex along the different rays.
Effective non-linear spinor dynamics in a spin-1 Bose-Einstein condensate
(2018-03) Michelangeli, Alessandro; Olgiati, Alessandro; Mathematics
We derive from first principles the experimentally observed effective dynamics of a spinor Bose gas initially prepared as a Bose–Einstein condensate and then left free to expand ballistically. In spinor condensates, which represent one of the recent frontiers in the manipulation of ultra-cold atoms, particles interact with a two-body spatial interaction and a spin–spin interaction. The effective dynamics is well-known to be governed by a system of coupled semi-linear Schrödinger equations: we recover this system, in the sense of marginals in the limit of infinitely many particles, with a mean-field re-scaling of the manybody Hamiltonian. When the resulting control of the dynamical persistence of condensation is quantified with the parameters of modern observations, we obtain a bound that remains quite accurate for the whole typical duration of the experiment.
Fractional powers and singular perturbations of quantum differential Hamiltonians
(SISSA, 2018-01) Michelangeli, Alessandro; Ottolini, Andrea; Scandone, Raffaele; Mathematics
We consider the fractional powers of singular (point-like) perturbations of the Laplacian and the singular perturbations of fractional powers of the Laplacian, and we compare two such constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schrödinger equations for the corresponding operators, we outline a programme of relevant questions that deserve being investigated.
Friedrichs systems in a Hilbert space framework: solvability and multiplicity
(2017-04) Antonić, Nenad; Erceg, Marko; Michelangeli, Alessandro; Mathematics
The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems, and proving that any pair of abstract Friedrichs operators admits bijective extensions with a signed boundary map. Moreover, we provide suffcient and necessary conditions for existence of infinitely many such pairs of spaces, and by the universal operator extension theory (Grubb, 1968) we get a complete identification of all such pairs, which we illustrate on two concrete one-dimensional examples.
Geometric Confinement and Dynamical Transmission of a Quantum Particle in Grushin Cylinder
(SISSA, 2019) Gallone, Matteo; Michelangeli, Alessandro; Pozzoli, Eugenio
We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally deﬁned on an inﬁnite cylinder equipped with an incom-plete Riemannian metric of Grushin type, in the non-trivial class of metrics yielding an inﬁnite deﬁciency index. Such realisations are naturally interpreted as Hamiltonians governing the geometric conﬁnement of a Schr¨odinger quan-tum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterise all physically meaningful exten-sions qualiﬁed by explicit local boundary conditions at the singularity. Within our general classiﬁcation we retrieve those distinguished extensions previously identiﬁed in the recent literature, namely the most conﬁning and the most transmitting one.
Global well-posedness of the magnetic Hartree equation with non-Strichartz external fields
(SISSA, 2015) Michelangeli, Alessandro
We study the magnetic Hartree equation with external fields to which magnetic Strichartz estimates are not necessarily applicable. We characterise the appropriate notion of energy space and in such a space we prove the global well-posedness of the associated initial value problem by means of energy methods only.
Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials
(2017) Antonelli, Paolo; Michelangeli, Alessandro; Scandone, Raffaele; Mathematics
We prove the existence of weak solutions in the space of energy for a class of nonlinear Schrödinger equations in the presence of a external, rough, time-dependent 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 regularisation, 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.
Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates
(2017-04-03) Michelangeli, Alessandro; Olgiati, Alessandro; Mathematics
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.
Ground state energy of mixture of Bose gases
(2018-03) Michelangeli, Alessandro; Thanh Nam, Phan; Olgiati, Alessandro; Mathematics
We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number N becomes large. In the dilute regime, when the interaction potentials have the length scale of order O(N-1), we show that the leading order of the ground state energy is captured correctly by the Gross-Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross- Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is O(1), we are able to verify Bogoliubov's approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaption to the multi-component setting is non-trivial in various respects and the analysis will be presented in details
Hydrogenoid Spectra with Central Perturbations
(2018-08) Gallone, Matteo; Michelangeli, Alessandro; Mathematics
Through the Kreĭn-Višik-Birman extension scheme, unlike the previous classical analysis based on von Neumann's theory, we reproduce the construction and classification of all self-adjoint realisations of two intimately related models: the three-dimensional hydrogenoid-like Hamiltonians with singular perturbation supported at the centre (the nucleus), and the Schördinger operators on the halfline with Coulomb potentials centred at the origin. These two problems are technically equivalent, albeit sometimes treated by their own in the the literature. Based on such scheme, we then recover the formula to determine the eigenvalues of each self-adjoint extension, which are corrections to the non-relativistic hydrogenoid energy levels.We discuss in which respect the Kreĭn-Višik-Birman scheme is somehow more natural in yielding the typical boundary condition of self-adjointness at the centre of the perturbation and in identifying the eigenvalues of each extension.
Krein-Vishik-Birman self-adjoint extension theory revisited
(2015-12-02) Michelangeli, Alessandro; Mathematics
The core results of the so-called Krein-Vishik-Birman theory of self-adjoint extensions of semi-bounded symmetric operators are reproduced, both in their original and in a more modern formulation, within a comprehensive discussion that includes missing details, elucidative steps, and intermediate results of independent interest.
Krein-Visik-Birman self-adjoint extension theory revisited
(2017) Gallone, Matteo; Michelangeli, Alessandro; Ottolini, Andrea; Mathematics
The core results of the so-called KreIn-Visik-Birman theory of self-adjoint extensions of semi-bounded symmetric operators are reproduced, both in their original and in a more modern formulation, within a comprehensive discussion that includes missing details, elucidative steps, and intermediate results of independent interest.
Krilov solvability of unbounded inverse linear problems
(SISSA, 2020-01-23) Caruso, Noe; Michelangeli, Alessandro
The abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem Af = g where A is a (possibly unbounded) linear operator on an inﬁnite-dimensional Hilbert space, and g is a datum in the range of A. The question consists of whether the solution f can be approximated in the Hilbert norm by ﬁnite linear combinations of g, Ag, A2g, . . . , and whether solutions of this sort exist and are unique. After revisiting the known picture when A is bounded, we study the general case of a densely deﬁned and closed A. Intrinsic operator-theoretic mechanisms are identiﬁed that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques
Krylov Solvability of Unbounded Inverse Linear Problems
(2019) Caruso, Noè Angelo; Michelangeli, Alessandro; mathematics
. The abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem Af = g where A is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and g is a datum in the range of A. The question consists of whether the solution f can be approximated in the Hilbert norm by finite linear combinations of g, Ag, A2g,... , and whether solutions of this sort exist and are unique. After revisiting the known picture when A is bounded, we study the general case of a densely defined and closed A. Intrinsic operator-theoretic mechanisms are identified that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques.
Lp-boundedness of wave operators for the three-dimensional multi-centre point interaction
(2017-04-13) Dell'Antonio, Gianfausto; Michelangeli, Alessandro; Scandone, Raffaele; Yajima, Kenji; Mathematics
We prove that, for arbitrary centres and strengths, the wave operators for three dimensional Schrödinger operators with multi-centre local point interactions are bounded in Lp(R3) for 1 < p < 3 and unbounded otherwise.

Browsing by Author "Michelangeli, Alessandro"

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