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.

1 aMichelangeli, Alessandro1 aNam, Phan, Thanh1 aOlgiati, Alessandro uhttps://doi.org/10.1142/S0129055X1950005301079nas a2200133 4500008004100000022001400041245009200055210006900147260000800216520062200224100002900846700002300875856004700898 2019 eng d a1661-826200aPoint-Like Perturbed Fractional Laplacians Through Shrinking Potentials of Finite Range0 aPointLike Perturbed Fractional Laplacians Through Shrinking Pote cMay3 aWe 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.

1 aMichelangeli, Alessandro1 aScandone, Raffaele uhttps://doi.org/10.1007/s11785-019-00927-w01371nas a2200145 4500008004100000245008000041210007100121260002400192300001100216490000700227520088900234100002901123700002401152856004901176 2018 eng d00aEffective non-linear spinor dynamics in a spin-1 Bose–Einstein condensate0 aEffective nonlinear spinor dynamics in a spin1 Bose–Einstein con bIOP Publishingcsep a4052010 v513 aWe 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.

1 aMichelangeli, Alessandro1 aOlgiati, Alessandro uhttps://doi.org/10.1088%2F1751-8121%2Faadbc200971nas a2200145 4500008004100000245008600041210006900127300001100196490000700207520050000214100002900714700002100743700002300764856003800787 2018 eng d00aFractional powers and singular perturbations of quantum differential Hamiltonians0 aFractional powers and singular perturbations of quantum differen a0721060 v593 aWe 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.

1 aMichelangeli, Alessandro1 aOttolini, Andrea1 aScandone, Raffaele uhttps://doi.org/10.1063/1.503385601226nas a2200193 4500008004100000022001400041245006800055210006500123300001600188490000800204520054000212653002300752653006700775653004400842100002300886700002900909700002300938856007100961 2018 eng d a0022-123600aOn fractional powers of singular perturbations of the Laplacian0 afractional powers of singular perturbations of the Laplacian a1551 - 16020 v2753 aWe 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.

10aPoint interactions10aRegular and singular component of a point-interaction operator10aSingular perturbations of the Laplacian1 aGeorgiev, Vladimir1 aMichelangeli, Alessandro1 aScandone, Raffaele uhttp://www.sciencedirect.com/science/article/pii/S002212361830104600800nas a2200121 4500008004100000245006300041210005900104520039700163100002000560700002900580700002100609856004800630 2018 en d00aOn Geometric Quantum Confinement in Grushin-Like Manifolds0 aGeometric Quantum Confinement in GrushinLike Manifolds3 aWe 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.1 aGallone, Matteo1 aMichelangeli, Alessandro1 aPozzoli, Eugenio uhttp://preprints.sissa.it/handle/1963/3532201283nas a2200169 4500008004100000022001400041245010100055210006900156260000800225300000700233490000700240520074700247100002100994700002901015700002301044856004601067 2018 eng d a1420-903900aGlobal, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials0 aGlobal finite energy weak solutions for the NLS with rough timed cMar a460 v693 aWe 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.

1 aAntonelli, Paolo1 aMichelangeli, Alessandro1 aScandone, Raffaele uhttps://doi.org/10.1007/s00033-018-0938-501340nas a2200109 4500008004100000245005100041210005100092520099000143100002001133700002901153856004801182 2018 en d00aHydrogenoid Spectra with Central Perturbations0 aHydrogenoid Spectra with Central Perturbations3 aThrough 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.1 aGallone, Matteo1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3532100840nas a2200133 4500008004100000245007200041210006800113260001000181520040400191100001600595700002900611700001800640856004800658 2018 en d00aOn Krylov solutions to infinite-dimensional inverse linear problems0 aKrylov solutions to infinitedimensional inverse linear problems bSISSA3 aWe 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.1 aCaruso, Noe1 aMichelangeli, Alessandro1 aNovati, Paolo uhttp://preprints.sissa.it/handle/1963/3532700806nas a2200181 4500008004100000022001400041245009400055210006900149260000800218300001400226490000700240520023200247100002900479700002900508700002300537700001800560856004600578 2018 eng d a1424-066100aLp-Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction0 aLpBoundedness of Wave Operators for the ThreeDimensional MultiCe cJan a283–3220 v193 aWe 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.

1 aDell'Antonio, Gianfausto1 aMichelangeli, Alessandro1 aScandone, Raffaele1 aYajima, Kenji uhttps://doi.org/10.1007/s00023-017-0628-400730nas a2200109 4500008004100000245008900041210006900130520032300199100002900522700002100551856004800572 2018 en d00aNon-linear Gross-Pitaevskii dynamics of a 2D binary condensate: a numerical analysis0 aNonlinear GrossPitaevskii dynamics of a 2D binary condensate a n3 aWe 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.1 aMichelangeli, Alessandro1 aPitton, Giuseppe uhttp://preprints.sissa.it/handle/1963/3532300448nas a2200097 4500008004100000245008100041210006900122100002900191700002300220856010700243 2018 eng d00aOn real resonances for the three-dimensional, multi-centre point interaction0 areal resonances for the threedimensional multicentre point inter1 aMichelangeli, Alessandro1 aScandone, Raffaele uhttps://www.math.sissa.it/publication/real-resonances-three-dimensional-multi-centre-point-interaction01292nas a2200157 4500008004100000245006900041210006900110260002100179300001200200490000700212520078900219100002901008700002401037700002301061856005001084 2018 eng d00aSingular Hartree equation in fractional perturbed Sobolev spaces0 aSingular Hartree equation in fractional perturbed Sobolev spaces bTaylor & Francis a558-5880 v253 aWe 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.

1 aMichelangeli, Alessandro1 aOlgiati, Alessandro1 aScandone, Raffaele uhttps://doi.org/10.1080/14029251.2018.150342301137nas a2200133 4500008004100000245009100041210006900132260001000201520068100211100001600892700002900908700001800937856004800955 2018 en d00aTruncation and convergence issues for bounded linear inverse problems in Hilbert space0 aTruncation and convergence issues for bounded linear inverse pro bSISSA3 aWe 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.1 aCaruso, Noe1 aMichelangeli, Alessandro1 aNovati, Paolo uhttp://preprints.sissa.it/handle/1963/3532600871nas a2200109 4500008004100000245005900041210005600100520051100156100001700667700002900684856004800713 2017 en d00aOn contact interactions realised as Friedrichs systems0 acontact interactions realised as Friedrichs systems3 aWe 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.1 aErceg, Marko1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3529801132nas a2200109 4500008004100000245006100041210006000102520076300162100002000925700002900945856004800974 2017 en d00aDiscrete spectra for critical Dirac-Coulomb Hamiltonians0 aDiscrete spectra for critical DiracCoulomb Hamiltonians3 aThe 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.1 aGallone, Matteo1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3530001416nas a2200169 4500008004100000020002200041245008400063210007000147260004400217300001400261520082100275100002001096700002301116700002901139700002901168856004901197 2017 eng d a978-3-319-58904-600aDispersive Estimates for Schrödinger Operators with Point Interactions in ℝ30 aDispersive Estimates for Schrödinger Operators with Point Intera aChambSpringer International Publishing a187–1993 aThe 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 $q \geq 3$.

1 aIandoli, Felice1 aScandone, Raffaele1 aMichelangeli, Alessandro1 aDell'Antonio, Gianfausto uhttps://doi.org/10.1007/978-3-319-58904-6_1101139nas a2200157 4500008004100000020002200041245007400063210006900137260004400206300001400250520058600264100002400850700002900874700002900903856004900932 2017 eng d a978-3-319-58904-600aEffective Non-linear Dynamics of Binary Condensates and Open Problems0 aEffective Nonlinear Dynamics of Binary Condensates and Open Prob aChambSpringer International Publishing a239–2563 aWe 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.

1 aOlgiati, Alessandro1 aMichelangeli, Alessandro1 aDell'Antonio, Gianfausto uhttps://doi.org/10.1007/978-3-319-58904-6_1401281nas a2200121 4500008004100000245008200041210006900123520085300192100002001045700001701065700002901082856004801111 2017 en d00aFriedrichs systems in a Hilbert space framework: solvability and multiplicity0 aFriedrichs systems in a Hilbert space framework solvability and 3 aThe 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.1 aAntonić, Nenad1 aErceg, Marko1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3528001947nas a2200145 4500008004100000245007100041210006800112260002100180300001200201490000700213520147800220100002901698700002401727856005001751 2017 eng d00aGross-Pitaevskii non-linear dynamics for pseudo-spinor condensates0 aGrossPitaevskii nonlinear dynamics for pseudospinor condensates bTaylor & Francis a426-4640 v243 aWe 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.

1 aMichelangeli, Alessandro1 aOlgiati, Alessandro uhttps://doi.org/10.1080/14029251.2017.134634800742nas a2200121 4500008004100000245006300041210006000104520033800164100002000502700002900522700002100551856004800572 2017 en d00aKrein-Visik-Birman self-adjoint extension theory revisited0 aKreinVisikBirman selfadjoint extension theory revisited3 aThe 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.1 aGallone, Matteo1 aMichelangeli, Alessandro1 aOttolini, Andrea uhttp://preprints.sissa.it/handle/1963/3528600961nas a2200157 4500008004100000022001400041245007700055210007100132260000800203300001400211490000600225520047300231100002900704700002400733856004600757 2017 eng d a1664-235X00aMean-field quantum dynamics for a mixture of Bose–Einstein condensates0 aMeanfield quantum dynamics for a mixture of Bose–Einstein conden cDec a377–4160 v73 aWe 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.

1 aMichelangeli, Alessandro1 aOlgiati, Alessandro uhttps://doi.org/10.1007/s13324-016-0147-300917nas a2200157 4500008004100000020002200041245008300063210006900146260004400215300001400259520035500273100002400628700002900652700002900681856004900710 2017 eng d a978-3-319-58904-600aRemarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian0 aRemarks on the Derivation of GrossPitaevskii Equation with Magne aChambSpringer International Publishing a257–2663 aThe 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.

1 aOlgiati, Alessandro1 aMichelangeli, Alessandro1 aDell'Antonio, Gianfausto uhttps://doi.org/10.1007/978-3-319-58904-6_1501120nas a2200109 4500008004100000245008000041210006900121520072300190100002000913700002900933856004800962 2017 en d00aSelf-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei0 aSelfadjoint realisations of the DiracCoulomb Hamiltonian for hea3 aWe 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.1 aGallone, Matteo1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3528701253nas a2200133 4500008004100000245007800041210006900119260001000188520080500198100001801003700002901021700002101050856004801071 2017 en d00aSpectral Properties of the 2+1 Fermionic Trimer with Contact Interactions0 aSpectral Properties of the 21 Fermionic Trimer with Contact Inte bSISSA3 aWe 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.1 aBecker, Simon1 aMichelangeli, Alessandro1 aOttolini, Andrea uhttp://preprints.sissa.it/handle/1963/3530301319nas a2200109 4500008004100000245011100041210006900152520088700221100002901108700002101137856005101158 2016 en d00aMultiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan--Skornyakov type0 aMultiplicity of selfadjoint realisations of the 21fermionic mode3 aWe 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.1 aMichelangeli, Alessandro1 aOttolini, Andrea uhttp://urania.sissa.it/xmlui/handle/1963/3526701002nas a2200109 4500008004100000245009900041210007000140520058100210100002900791700002100820856005100841 2016 en d00aNon-linear Schrödinger system for the dynamics of a binary condensate: theory and 2D numerics0 aNonlinear Schrödinger system for the dynamics of a binary conden3 aWe 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.1 aMichelangeli, Alessandro1 aPitton, Giuseppe uhttp://urania.sissa.it/xmlui/handle/1963/3526601116nas a2200109 4500008004100000245007800041210006900119520071700188100002900905700002100934856005100955 2016 en d00aOn point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians0 apoint interactions realised as TerMartirosyanSkornyakov Hamilton3 aFor 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.1 aMichelangeli, Alessandro1 aOttolini, Andrea uhttp://urania.sissa.it/xmlui/handle/1963/3519501837nas a2200145 4500008004100000245007900041210006900120520133000189100002201519700002901541700002001570700002901590700002101619856005101640 2015 en d00aA class of Hamiltonians for a three-particle fermionic system at unitarity0 aclass of Hamiltonians for a threeparticle fermionic system at un3 aWe 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.1 aCorreggi, Michele1 aDell'Antonio, Gianfausto1 aFinco, Domenico1 aMichelangeli, Alessandro1 aTeta, Alessandro uhttp://urania.sissa.it/xmlui/handle/1963/3446900723nas a2200109 4500008004100000245009500041210006900136260001000205520031800215100002900533856005100562 2015 en d00aGlobal well-posedness of the magnetic Hartree equation with non-Strichartz external fields0 aGlobal wellposedness of the magnetic Hartree equation with nonSt bSISSA3 aWe 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.1 aMichelangeli, Alessandro uhttp://urania.sissa.it/xmlui/handle/1963/3444001018nas a2200121 4500008004100000245007900041210007000120260001000190520058700200100002900787700002900816856005100845 2015 en d00aSchödinger operators on half-line with shrinking potentials at the origin0 aSchödinger operators on halfline with shrinking potentials at th bSISSA3 aWe 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.1 aDell'Antonio, Gianfausto1 aMichelangeli, Alessandro uhttp://urania.sissa.it/xmlui/handle/1963/3443901282nas a2200121 4500008004100000245007500041210006900116260001000185520086400195100002901059700002101088856005101109 2015 en d00aStability of closed gaps for the alternating Kronig-Penney Hamiltonian0 aStability of closed gaps for the alternating KronigPenney Hamilt bSISSA3 aWe 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.1 aMichelangeli, Alessandro1 aMonaco, Domenico uhttp://urania.sissa.it/xmlui/handle/1963/3446001228nas a2200109 4500008004100000245007200041210006700113520083900180100002901019700001901048856005101067 2015 en d00aStability of the (2+2)-fermionic system with zero-range interaction0 aStability of the 22fermionic system with zerorange interaction3 aWe 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.1 aMichelangeli, Alessandro1 aPfeiffer, Paul uhttp://urania.sissa.it/xmlui/handle/1963/3447400787nas a2200121 4500008004100000245013000041210006900171260001000240520031200250100002300562700002900585856005100614 2015 en d00aTranslation and adaptation of Birman's paper "On the theory of self-adjoint extensions of positive definite operators" (1956)0 aTranslation and adaptation of Birmans paper On the theory of sel bSISSA3 aThis 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.1 aKhotyakov, Mikhail1 aMichelangeli, Alessandro uhttp://urania.sissa.it/xmlui/handle/1963/3444301107nas a2200121 4500008004300000245007000043210006800113260001000181520068600191100002900877700002900906856005000935 2014 en_Ud 00aDynamics on a graph as the limit of the dynamics on a "fat graph"0 aDynamics on a graph as the limit of the dynamics on a fat graph bSISSA3 aWe 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.1 aDell'Antonio, Gianfausto1 aMichelangeli, Alessandro uhttp://urania.sissa.it/xmlui/handle/1963/748501607nas a2200157 4500008004100000245009600041210006900137260002100206520106500227100002201292700002901314700002001343700002901363700002101392856003601413 2012 en d00aStability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions0 aStability for a System of N Fermions Plus a Different Particle w bWorld Scientific3 aWe 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.1 aCorreggi, Michele1 aDell'Antonio, Gianfausto1 aFinco, Domenico1 aMichelangeli, Alessandro1 aTeta, Alessandro uhttp://hdl.handle.net/1963/606901174nas a2200109 4500008004300000245005400043210005300097520082800150100002900978700002101007856003601028 2009 en_Ud 00a1D periodic potentials with gaps vanishing at k=00 a1D periodic potentials with gaps vanishing at k03 aAppearance 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.1 aMichelangeli, Alessandro1 aZagordi, Osvaldo uhttp://hdl.handle.net/1963/181800767nas a2200097 4500008004300000245005300043210004900096520045900145100002900604856003600633 2008 en_Ud 00aEquivalent definitions of asymptotic 100% B.E.C.0 aEquivalent definitions of asymptotic 100 BEC3 aIn 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.1 aMichelangeli, Alessandro uhttp://hdl.handle.net/1963/254600337nas a2200085 4500008004300000245007400043210006900117100002900186856003600215 2007 en_Ud 00aBose-Einstein condensation: analysis of problems and rigorous results0 aBoseEinstein condensation analysis of problems and rigorous resu1 aMichelangeli, Alessandro uhttp://hdl.handle.net/1963/218901299nas a2200097 4500008004300000245006000043210005900103520097400162100002901136856003601165 2007 en_Ud 00aReduced density matrices and Bose-Einstein condensation0 aReduced density matrices and BoseEinstein condensation3 aEmergence 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.1 aMichelangeli, Alessandro uhttp://hdl.handle.net/1963/198600981nas a2200097 4500008004300000245008200043210006900125520062400194100002900818856003600847 2007 en_Ud 00aRole of scaling limits in the rigorous analysis of Bose-Einstein condensation0 aRole of scaling limits in the rigorous analysis of BoseEinstein 3 aIn 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.1 aMichelangeli, Alessandro uhttp://hdl.handle.net/1963/198400738nas a2200097 4500008004300000245008700043210006900130520037600199100002900575856003600604 2007 en_Ud 00aStrengthened convergence of marginals to the cubic nonlinear Schroedinger equation0 aStrengthened convergence of marginals to the cubic nonlinear Sch3 aWe 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.1 aMichelangeli, Alessandro uhttp://hdl.handle.net/1963/197701209nas a2200097 4500008004300000245009800043210006900141520083600210100002901046856003601075 2006 en_Ud 00aBorn approximation in the problem of the rigorous derivation of the Gross-Pitaevskii equation0 aBorn approximation in the problem of the rigorous derivation of 3 a\\\"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 limit1 aMichelangeli, Alessandro uhttp://hdl.handle.net/1963/1819