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 adaptation to the multi-component setting is non-trivial in various respects and the analysis will be presented in detail.

%B Reviews in Mathematical Physics %V 31 %P 1950005 %G eng %U https://doi.org/10.1142/S0129055X19500053 %R 10.1142/S0129055X19500053 %0 Journal Article %J Complex Analysis and Operator Theory %D 2019 %T Point-Like Perturbed Fractional Laplacians Through Shrinking Potentials of Finite Range %A Alessandro Michelangeli %A Raffaele Scandone %XWe construct the rank-one, singular (point-like) perturbations of the d-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schrödinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for Schrödinger operators formed by a fractional Laplacian and a regular potential.

%B Complex Analysis and Operator Theory %8 May %G eng %U https://doi.org/10.1007/s11785-019-00927-w %R 10.1007/s11785-019-00927-w %0 Journal Article %J Journal of Physics A: Mathematical and Theoretical %D 2018 %T Effective non-linear spinor dynamics in a spin-1 Bose–Einstein condensate %A Alessandro Michelangeli %A Alessandro Olgiati %XWe 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 many-body 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.

%B Journal of Physics A: Mathematical and Theoretical %I IOP Publishing %V 51 %P 405201 %8 sep %G eng %U https://doi.org/10.1088%2F1751-8121%2Faadbc2 %R 10.1088/1751-8121/aadbc2 %0 Journal Article %J Journal of Mathematical Physics %D 2018 %T Fractional powers and singular perturbations of quantum differential Hamiltonians %A Alessandro Michelangeli %A Andrea Ottolini %A Raffaele Scandone %XWe 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.

%B Journal of Mathematical Physics %V 59 %P 072106 %G eng %U https://doi.org/10.1063/1.5033856 %R 10.1063/1.5033856 %0 Journal Article %J Journal of Functional Analysis %D 2018 %T On fractional powers of singular perturbations of the Laplacian %A Vladimir Georgiev %A Alessandro Michelangeli %A Raffaele Scandone %K Point interactions %K Regular and singular component of a point-interaction operator %K Singular perturbations of the Laplacian %XWe 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.

%B Journal of Functional Analysis %V 275 %P 1551 - 1602 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022123618301046 %R https://doi.org/10.1016/j.jfa.2018.03.007 %0 Report %D 2018 %T On Geometric Quantum Confinement in Grushin-Like Manifolds %A Matteo Gallone %A Alessandro Michelangeli %A Eugenio Pozzoli %X We study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl's analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace-Beltrami operator. %G en %U http://preprints.sissa.it/handle/1963/35322 %1 35632 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-09-19T07:05:15Z No. of bitstreams: 1 GMP-Grushin-SISSApreprint.pdf: 390608 bytes, checksum: c4bbb299a3b07668840d185c315c1a29 (MD5) %0 Journal Article %J Zeitschrift für angewandte Mathematik und Physik %D 2018 %T Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials %A Paolo Antonelli %A Alessandro Michelangeli %A Raffaele Scandone %XWe 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.

%B Zeitschrift für angewandte Mathematik und Physik %V 69 %P 46 %8 Mar %G eng %U https://doi.org/10.1007/s00033-018-0938-5 %R 10.1007/s00033-018-0938-5 %0 Report %D 2018 %T Hydrogenoid Spectra with Central Perturbations %A Matteo Gallone %A Alessandro Michelangeli %X 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. %G en %U http://preprints.sissa.it/handle/1963/35321 %1 35631 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-08-27T06:42:30Z No. of bitstreams: 1 GM-hydrogenoid-2018-SISSA-Preprint.pdf: 480482 bytes, checksum: 228ebe556a2688a43dcbbd1edeeaa5c4 (MD5) %0 Report %D 2018 %T On Krylov solutions to infinite-dimensional inverse linear problems %A Noe Caruso %A Alessandro Michelangeli %A Paolo Novati %X We discuss, in the context of inverse linear problems in Hilbert space, the notion of the associated infinite-dimensional Krylov subspace and we produce necessary and sufficient conditions for the Krylov-solvability of the considered inverse problem. The presentation is based on theoretical results together with a series of model examples, and it is corroborated by specific numerical experiments. %I SISSA %G en %U http://preprints.sissa.it/handle/1963/35327 %1 35638 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-11-20T13:12:02Z No. of bitstreams: 1 krylov_InfiniteDim_sissapreprint.pdf: 472654 bytes, checksum: 54ffc66b6879cbb031453355bf6ed56b (MD5) %0 Journal Article %J Annales Henri Poincaré %D 2018 %T Lp-Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction %A Gianfausto Dell'Antonio %A Alessandro Michelangeli %A Raffaele Scandone %A Kenji Yajima %XWe 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.

%B Annales Henri Poincaré %V 19 %P 283–322 %8 Jan %G eng %U https://doi.org/10.1007/s00023-017-0628-4 %R 10.1007/s00023-017-0628-4 %0 Report %D 2018 %T Non-linear Gross-Pitaevskii dynamics of a 2D binary condensate: a numerical analysis %A Alessandro Michelangeli %A Giuseppe Pitton %X We present a numerical study of the two-dimensional Gross-Pitaevskii systems in a wide range of relevant regimes of population ratios and intra-species and inter-species interactions. Our numerical method is based on a Fourier collocation scheme in space combined with a fourth order integrating factor scheme in time. %G en %U http://preprints.sissa.it/handle/1963/35323 %1 35633 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-09-20T06:04:10Z No. of bitstreams: 1 SISSA_preprint_35-2018-MATE.pdf: 2536075 bytes, checksum: 323dca7431103028ecadfc71c052f4ed (MD5) %0 Report %D 2018 %T On real resonances for the three-dimensional, multi-centre point interaction %A Alessandro Michelangeli %A Raffaele Scandone %G eng %0 Journal Article %J Journal of Nonlinear Mathematical Physics %D 2018 %T Singular Hartree equation in fractional perturbed Sobolev spaces %A Alessandro Michelangeli %A Alessandro Olgiati %A Raffaele Scandone %XWe establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.

%B Journal of Nonlinear Mathematical Physics %I Taylor & Francis %V 25 %P 558-588 %G eng %U https://doi.org/10.1080/14029251.2018.1503423 %R 10.1080/14029251.2018.1503423 %0 Report %D 2018 %T Truncation and convergence issues for bounded linear inverse problems in Hilbert space %A Noe Caruso %A Alessandro Michelangeli %A Paolo Novati %X We present a general discussion of the main features and issues that (bounded) inverse linear problems in Hilbert space exhibit when the dimension of the space is infinite. This includes the set-up of a consistent notation for inverse problems that are genuinely infinite-dimensional, the analysis of the finite-dimensional truncations, a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests. %I SISSA %G en %U http://preprints.sissa.it/handle/1963/35326 %1 35637 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-11-20T13:02:23Z No. of bitstreams: 1 infinite_dim_truncation_sissapreprint.pdf: 1326956 bytes, checksum: 75a6af69b0bca0c5b9b5283e640d89be (MD5) %0 Report %D 2017 %T On contact interactions realised as Friedrichs systems %A Marko Erceg %A Alessandro Michelangeli %X We realise the Hamiltonians of contact interactions in quantum mechanics within the framework of abstract Friedrichs systems. In particular, we show that the construction of the self-adjoint (or even only closed) operators of contact interaction supported at a fixed point can be associated with the construction of the bijective realisations of a suitable pair of abstract Friedrich operators. In this respect, the Hamiltonians of contact interaction provide novel examples of abstract Friedrich systems. %G en %U http://preprints.sissa.it/handle/1963/35298 %1 35604 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-10-10T11:23:22Z No. of bitstreams: 1 SISSA_preprint_48-2017-MATE.pdf: 431531 bytes, checksum: 0368c692a322778cda197c49c320b3a6 (MD5) %0 Report %D 2017 %T Discrete spectra for critical Dirac-Coulomb Hamiltonians %A Matteo Gallone %A Alessandro Michelangeli %X 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. %G en %U http://preprints.sissa.it/handle/1963/35300 %1 35606 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-11-06T14:27:38Z No. of bitstreams: 1 SISSA_preprint_44-2017-MATE.pdf: 1573857 bytes, checksum: 728e94322471701d89ed5741cda85d99 (MD5) %0 Book Section %B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %D 2017 %T Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3 %A Felice Iandoli %A Raffaele Scandone %E Alessandro Michelangeli %E Gianfausto Dell'Antonio %XThe study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p$−$L^q$ estimates of the free Laplacian in the smaller regime $q \in [2,3)$. These estimates are implied by a recent result concerning the Lpboundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q \geq 3$.

%B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %I Springer International Publishing %C Cham %P 187–199 %@ 978-3-319-58904-6 %G eng %U https://doi.org/10.1007/978-3-319-58904-6_11 %R 10.1007/978-3-319-58904-6_11 %0 Book Section %B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %D 2017 %T Effective Non-linear Dynamics of Binary Condensates and Open Problems %A Alessandro Olgiati %E Alessandro Michelangeli %E Gianfausto Dell'Antonio %XWe report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Schödinger equations. After reviewing and commenting our proof in the mean-field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.

%B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %I Springer International Publishing %C Cham %P 239–256 %@ 978-3-319-58904-6 %G eng %U https://doi.org/10.1007/978-3-319-58904-6_14 %R 10.1007/978-3-319-58904-6_14 %0 Report %D 2017 %T Friedrichs systems in a Hilbert space framework: solvability and multiplicity %A Nenad Antonić %A Marko Erceg %A Alessandro Michelangeli %X 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. %G en %U http://preprints.sissa.it/handle/1963/35280 %1 35587 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-04-11T07:42:37Z No. of bitstreams: 1 SISSA_preprint_16-2017-MATE.pdf: 323262 bytes, checksum: 17892409d83085ec46707fac64056fc1 (MD5) %0 Journal Article %J Journal of Nonlinear Mathematical Physics %D 2017 %T Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates %A Alessandro Michelangeli %A Alessandro Olgiati %XWe 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.

%B Journal of Nonlinear Mathematical Physics %I Taylor & Francis %V 24 %P 426-464 %G eng %U https://doi.org/10.1080/14029251.2017.1346348 %R 10.1080/14029251.2017.1346348 %0 Report %D 2017 %T Krein-Visik-Birman self-adjoint extension theory revisited %A Matteo Gallone %A Alessandro Michelangeli %A Andrea Ottolini %X 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. %G en %U http://preprints.sissa.it/handle/1963/35286 %1 35591 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-05-31T08:18:47Z No. of bitstreams: 1 Gallone-Michelangeli-Ottolini-KVB.pdf: 675862 bytes, checksum: 325168fcaae89a9faedde0e5c32e69a7 (MD5) %0 Journal Article %J Analysis and Mathematical Physics %D 2017 %T Mean-field quantum dynamics for a mixture of Bose–Einstein condensates %A Alessandro Michelangeli %A Alessandro Olgiati %XWe 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.

%B Analysis and Mathematical Physics %V 7 %P 377–416 %8 Dec %G eng %U https://doi.org/10.1007/s13324-016-0147-3 %R 10.1007/s13324-016-0147-3 %0 Book Section %B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %D 2017 %T Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian %A Alessandro Olgiati %E Alessandro Michelangeli %E Gianfausto Dell'Antonio %XThe effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the ``projection counting'' scheme.

%B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %I Springer International Publishing %C Cham %P 257–266 %@ 978-3-319-58904-6 %G eng %U https://doi.org/10.1007/978-3-319-58904-6_15 %R 10.1007/978-3-319-58904-6_15 %0 Report %D 2017 %T Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei %A Matteo Gallone %A Alessandro Michelangeli %X We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the KreĬn-Višik- Birman extension scheme, or also on Grubb's universal classification theory, as opposite to previous works within the standard von Neu- mann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the exten- sion, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension. %G en %U http://preprints.sissa.it/handle/1963/35287 %1 35592 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-06-05T06:42:39Z No. of bitstreams: 1 Coulomb_Gallone_Michelangeli_26_2017.pdf: 508858 bytes, checksum: 21c788d0de4fed88bee2e6fd5cd0849c (MD5) %0 Journal Article %D 2017 %T Spectral Properties of the 2+1 Fermionic Trimer with Contact Interactions %A Simon Becker %A Alessandro Michelangeli %A Andrea Ottolini %X 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 and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses. %I SISSA %G en %U http://preprints.sissa.it/handle/1963/35303 %1 35609 %2 Mathematics %4 1 %$ Submitted by mmarin@sissa.it (mmarin@sissa.it) on 2018-01-04T08:57:37Z No. of bitstreams: 1 SISSA_preprint_61-2017-MATE.pdf: 465833 bytes, checksum: bc2bbd7578ccfcd69ca2a33a2fc3adb0 (MD5) %0 Report %D 2016 %T Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan--Skornyakov type %A Alessandro Michelangeli %A Andrea Ottolini %X 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 Kreĭn, Višiik, and Birman. We identify the explicit `Kreĭn-Višik-Birman extension param- eter' as an operator on the `space of charges' for this model (the `Kreĭn 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 re- produce for the first time the previous partial constructions obtained by means of an alternative quadratic form approach. %G en %U http://urania.sissa.it/xmlui/handle/1963/35267 %1 35573 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-01-16T12:25:08Z No. of bitstreams: 1 SISSA_preprint_65-2016-MATE.pdf: 252490 bytes, checksum: 59617b8bff76543e4ea137b885bf1254 (MD5) %0 Report %D 2016 %T Non-linear Schrödinger system for the dynamics of a binary condensate: theory and 2D numerics %A Alessandro Michelangeli %A Giuseppe Pitton %X We present a comprehensive discussion of the mathematical framework for binary Bose-Einstein condensates and for the rigorous derivation of their effective dynamics, governed by a system of coupled non-linear Gross-Pitaevskii equations. We also develop in the 2D case a systematic numerical study of the Gross-Pitaevskii systems in a wide range of relevant regimes of population ratios and intra-species and inter-species interactions. Our numerical method is based on a Fourier collocation scheme in space combined with a fourth order integrating factor scheme in time. %G en %U http://urania.sissa.it/xmlui/handle/1963/35266 %1 35572 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-01-16T12:09:24Z No. of bitstreams: 1 SISSA_preprint_63-2016-MATE_Michelangeli-Pitton-2016.pdf: 6158349 bytes, checksum: ab11de2762ff510e6833474d0688a8b4 (MD5) %0 Report %D 2016 %T On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians %A Alessandro Michelangeli %A Andrea Ottolini %X 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 point-wise asymptotics, and we discuss the consequences of this fact on a model of two identical fermions and a third particle of different nature. %G en %U http://urania.sissa.it/xmlui/handle/1963/35195 %1 35489 %2 Mathematics %4 1 %# MAT/07 %$ Submitted by aottolini@sissa.it (aottolini@sissa.it) on 2016-06-16T11:35:16Z No. of bitstreams: 1 SISSA_preprint_11-2016-MATE.pdf: 288426 bytes, checksum: 1cd4fd0554e316274d2160eaecb2646e (MD5) %0 Report %D 2015 %T A class of Hamiltonians for a three-particle fermionic system at unitarity %A Michele Correggi %A Gianfausto Dell'Antonio %A Domenico Finco %A Alessandro Michelangeli %A Alessandro Teta %X 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. %G en %U http://urania.sissa.it/xmlui/handle/1963/34469 %1 34644 %2 Mathematics %4 1 %$ Submitted by Alessandro Michelangeli (alemiche@sissa.it) on 2015-05-21T06:33:20Z No. of bitstreams: 1 SISSA_preprint_22-2015-MATE.pdf: 438261 bytes, checksum: ca05e5e2c5d78d87225a11073cb09d47 (MD5) %0 Report %D 2015 %T Global well-posedness of the magnetic Hartree equation with non-Strichartz external fields %A Alessandro Michelangeli %X 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. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/34440 %1 34567 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-02-09T07:36:23Z No. of bitstreams: 1 SISSA_preprint_07-2015-MATE.pdf: 185909 bytes, checksum: ae723e94b4951f0092f096c72ebf14d6 (MD5) %0 Report %D 2015 %T Schödinger operators on half-line with shrinking potentials at the origin %A Gianfausto Dell'Antonio %A Alessandro Michelangeli %X We discuss the general model of a Schrödinger quantum particle constrained on a straight half-line with given self-adjoint boundary condition at the origin and an interaction potential supported around the origin. We study the limit when the range of the potential scales to zero and its magnitude blows up. We show that in the limit the dynamics is generated by a self-adjoint negative Laplacian on the half-line, with a possible preservation or modification of the boundary condition at the origin, depending on the magnitude of the scaling and of the strength of the potential. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/34439 %1 34566 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-02-09T07:28:40Z No. of bitstreams: 1 SISSA_preprint_06-2015-MATE.pdf: 177243 bytes, checksum: 383296953d36b4cec0cb4cfb55e95d7f (MD5) %0 Report %D 2015 %T Stability of closed gaps for the alternating Kronig-Penney Hamiltonian %A Alessandro Michelangeli %A Domenico Monaco %X We consider the Kronig-Penney model for a quantum crystal with equispaced periodic delta-interactions of alternating strength. For this model all spectral gaps at the centre of the Brillouin zone are known to vanish, although so far this noticeable property has only been proved through a very delicate analysis of the discriminant of the corresponding ODE and the associated monodromy matrix. We provide a new, alternative proof by showing that this model can be approximated, in the norm resolvent sense, by a model of regular periodic interactions with finite range for which all gaps at the centre of the Brillouin zone are still vanishing. In particular this shows that the vanishing gap property is stable in the sense that it is present also for the "physical" approximants and is not only a feature of the idealised model of zero-range interactions. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/34460 %1 34629 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-04-01T14:57:03Z No. of bitstreams: 1 SISSA_preprint_16-2015-MATE.pdf: 349807 bytes, checksum: f5e6dacd85fb41c96420795cc37cc12c (MD5) %0 Report %D 2015 %T Stability of the (2+2)-fermionic system with zero-range interaction %A Alessandro Michelangeli %A Paul Pfeiffer %X We introduce a 3D model, and we study its stability, consisting of two distinct pairs of identical fermions coupled with a two-body interaction between fermions of different species, whose effective range is essentially zero (a so called (2+2)-fermionic system with zero-range interaction). The interaction is modelled by implementing the the celebrated (and ubiquitous, in the literature of this field) Bethe-Peierls contact condition with given two-body scattering length within the Krein-Visik-Birman theory of extensions of semi-bounded symmetric operators, in order to make the Hamiltonian a well-defined (self-adjoint) physical observable. After deriving the expression for the associated energy quadratic form, we show analytically and numerically that the energy of the model is bounded below, thus describing a stable system. %G en %U http://urania.sissa.it/xmlui/handle/1963/34474 %1 34649 %2 Mathematics %4 1 %# MAT/07 %$ Submitted by Alessandro Michelangeli (alemiche@sissa.it) on 2015-06-24T10:47:38Z No. of bitstreams: 1 sissa-preprint-29-2015-mate.pdf: 827153 bytes, checksum: e5268c2d26f348929a3c532f9ffd3097 (MD5) %0 Report %D 2015 %T Translation and adaptation of Birman's paper "On the theory of self-adjoint extensions of positive definite operators" (1956) %A Mikhail Khotyakov %A Alessandro Michelangeli %X This is an accurate translation from Russian and adaptation to the modern mathematical jargon of a classical paper by M. Sh. Birman published in 1956, which is still today central in the theory of self-adjoint extensions of semi-bounded operators, and for which yet no English version was available so far. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/34443 %1 34570 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-02-13T10:07:37Z No. of bitstreams: 1 SISSA_preprint_08-2015-MATE(1).pdf: 2044497 bytes, checksum: 092e3561b5722c604c896ccc97db7a76 (MD5) %0 Report %D 2014 %T Dynamics on a graph as the limit of the dynamics on a "fat graph" %A Gianfausto Dell'Antonio %A Alessandro Michelangeli %X 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. %I SISSA %G en_US %U http://urania.sissa.it/xmlui/handle/1963/7485 %1 7598 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-12-12T13:16:15Z No. of bitstreams: 1 sissa-preprint-69-2014-mate.pdf: 137454 bytes, checksum: 4d4571bd97df46ee020c81057ce78620 (MD5) %0 Journal Article %J Rev. Math. Phys. 24 (2012), 1250017 %D 2012 %T Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions %A Michele Correggi %A Gianfausto Dell'Antonio %A Domenico Finco %A Alessandro Michelangeli %A Alessandro Teta %X We study the stability problem for a non-relativistic quantum system in\\r\\ndimension three composed by $ N \\\\geq 2 $ identical fermions, with unit mass,\\r\\ninteracting with a different particle, with mass $ m $, via a zero-range\\r\\ninteraction of strength $ \\\\alpha \\\\in \\\\R $. We construct the corresponding\\r\\nrenormalised quadratic (or energy) form $ \\\\form $ and the so-called\\r\\nSkornyakov-Ter-Martirosyan symmetric extension $ H_{\\\\alpha} $, which is the\\r\\nnatural candidate as Hamiltonian of the system. We find a value of the mass $\\r\\nm^*(N) $ such that for $ m > m^*(N)$ the form $ \\\\form $ is closed and bounded from below. As a consequence, $ \\\\form $ defines a unique self-adjoint and bounded from below extension of $ H_{\\\\alpha}$ and therefore the system is stable. On the other hand, we also show that the form $ \\\\form $ is unbounded from below for $ m < m^*(2)$. In analogy with the well-known bosonic case, this suggests that the system is unstable for $ m < m^*(2)$ and the so-called Thomas effect occurs. %B Rev. Math. Phys. 24 (2012), 1250017 %I World Scientific %G en %U http://hdl.handle.net/1963/6069 %1 5955 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2012-08-02T07:19:59Z\\nNo. of bitstreams: 1\\n1201.5740v1.pdf: 303047 bytes, checksum: eb2df0afd547514c235422e82f494584 (MD5) %R 10.1142/S0129055X12500171 %0 Journal Article %J Mem. Differential Equations Math. Phys. 47 (2009) 133-158 %D 2009 %T 1D periodic potentials with gaps vanishing at k=0 %A Alessandro Michelangeli %A Osvaldo Zagordi %X Appearance of energy bands and gaps in the dispersion relations of a periodic potential is a standard feature of Quantum Mechanics. We investigate the class of one-dimensional periodic potentials for which all gaps vanish at the center of the Brillouin zone. We characterise themthrough a necessary and sufficient condition. Potentials of the form we focus on arise in different fields of Physics, from supersymmetric Quantum Mechanics, to Korteweg-de Vries equation theory and classical diffusion problems. The O.D.E. counterpart to this problem is the characterisation of periodic potentials for which coexistence occurs of linearly independent solutions of the corresponding Schrödinger equation (Hill\\\'s equation). This result is placed in the perspective of the previous related results available in the literature. %B Mem. Differential Equations Math. Phys. 47 (2009) 133-158 %G en_US %U http://hdl.handle.net/1963/1818 %1 2396 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-04-18T13:06:21Z\\nNo. of bitstreams: 1\\ncond-mat0603172.pdf: 227250 bytes, checksum: 8b182847d3be1ae07188b40594750f64 (MD5) %0 Journal Article %J Nuovo Cimento B 123 (2008) 181-192 %D 2008 %T Equivalent definitions of asymptotic 100% B.E.C. %A Alessandro Michelangeli %X In the mathematical analysis Bose-Einstein condensates, in particular in the study of the quantum dynamics, some kind of factorisation property has been recently proposed as a convenient technical assumption of condensation. After having surveyed both the standard definition of complete Bose-Einstein condensation in the limit of infinitely many particles and some forms of asymptotic factorisation, we prove that these characterisations are equivalent. %B Nuovo Cimento B 123 (2008) 181-192 %G en_US %U http://hdl.handle.net/1963/2546 %1 1573 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-01-10T13:15:49Z\\nNo. of bitstreams: 1\\nsissa_preprint102_07.pdf: 215355 bytes, checksum: a09c310627ed3ed9acf1156596e7dd7e (MD5) %R 10.1393/ncb/i2008-10521-y %0 Report %D 2007 %T Bose-Einstein condensation: analysis of problems and rigorous results %A Alessandro Michelangeli %G en_US %U http://hdl.handle.net/1963/2189 %1 2055 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-08T06:59:18Z\\nNo. of bitstreams: 1\\n70-2007MP.pdf: 1694233 bytes, checksum: 36e2ed8af8e03d149a0b11cccfd01634 (MD5) %0 Report %D 2007 %T Reduced density matrices and Bose-Einstein condensation %A Alessandro Michelangeli %X Emergence and applications of the ubiquitous tool of reduced density matrices in the rigorous analysis of Bose Einstein condensation is reviewed, and new related results are added. The need and the nature of scaling limits of infinitely many particles is discussed, which imposes that a physically meaningful and mathematically well-posed definition of asymptotic condensation is placed at the level of marginals.\\nThe topic of correlations in the condensed state is addressed in order to show their influence at this level of marginals, both in the true condensed state and in the suitable trial functions one introduces to approximate the many-body structure and energy. Complete condensation is shown to be equivalently defined at any fixed k-body level, both for pure and mixed states. Further, it is proven to be equivalent to some other characterizations in terms of asymptotic factorization of the many-body state, which are currently present in the literature. %G en_US %U http://hdl.handle.net/1963/1986 %1 2210 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-08-07T12:22:30Z\\nNo. of bitstreams: 1\\n39-2007MP.pdf: 563889 bytes, checksum: b6688b840128220a2e61def6a60ff7d9 (MD5) %0 Journal Article %J J. Math. Phys. 48 (2007) 102102 %D 2007 %T Role of scaling limits in the rigorous analysis of Bose-Einstein condensation %A Alessandro Michelangeli %X In the context of the rigorous analysis of Bose-Einstein condensation, recent achievements have been obtained in the form of asymptotic results when some appropriate scaling is performed in the Hamiltonian, and the limit of infinite number of particles is taken. In particular, two modified thermodynamic limits of infinite dilution turned out to provide an insight in this analysis, the so-\\ncalled Gross-Pitaevskii limit and the related Tomas-Fermi limit. Here such scalings are discussed with respect to their physical and mathematical motivations, and to the currently known results obtained within this framework. %B J. Math. Phys. 48 (2007) 102102 %G en_US %U http://hdl.handle.net/1963/1984 %1 2212 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-07-26T14:30:55Z\\nNo. of bitstreams: 1\\n14-2007-MP.pdf: 485881 bytes, checksum: 3cf7764eded7853d876ad4793569bdfa (MD5) %R 10.1063/1.2783114 %0 Report %D 2007 %T Strengthened convergence of marginals to the cubic nonlinear Schroedinger equation %A Alessandro Michelangeli %X We rewrite a recent derivation of the cubic non-linear Schroedinger equation by Adami, Golse, and Teta in the more natural formof the asymptotic factorisation of marginals at any fixed time and in the trace norm. This is the standard form in which the emergence of the\\nnon-linear effective dynamics of a large system of interacting bosons is\\nproved in the literature. %G en_US %U http://hdl.handle.net/1963/1977 %1 2218 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-06-06T09:21:56Z\\nNo. of bitstreams: 1\\nstrengthened.pdf: 228368 bytes, checksum: c87b1b3ae8fd722412cf906c5f44df5b (MD5) %0 Report %D 2006 %T Born approximation in the problem of the rigorous derivation of the Gross-Pitaevskii equation %A Alessandro Michelangeli %X \\\"It has a flavour of Mathematical Physics...\\\"With these words, just few years ago, prof. Di Giacomo\\nused 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\\nthe 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 %G en_US %U http://hdl.handle.net/1963/1819 %1 2395 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-04-18T13:24:51Z\\nNo. of bitstreams: 1\\nSISSA13-2006FM.pdf: 154961 bytes, checksum: 6a952a2dc9b448e8b3476560ad108a93 (MD5)