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.

VL - 31 UR - https://doi.org/10.1142/S0129055X19500053 ER - TY - JOUR T1 - Cohesive fracture with irreversibility: Quasistatic evolution for a model subject to fatigue JF - Mathematical Models and Methods in Applied Sciences Y1 - 2018 A1 - Vito Crismale A1 - Giuliano Lazzaroni A1 - Gianluca Orlando AB -In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e. a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement.

VL - 28 UR - https://doi.org/10.1142/S0218202518500379 ER - TY - JOUR T1 - Effective non-linear spinor dynamics in a spin-1 Bose–Einstein condensate JF - Journal of Physics A: Mathematical and Theoretical Y1 - 2018 A1 - Alessandro Michelangeli A1 - Alessandro Olgiati AB -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 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.

PB - IOP Publishing VL - 51 UR - https://doi.org/10.1088%2F1751-8121%2Faadbc2 ER - TY - JOUR T1 - Fractional powers and singular perturbations of quantum differential Hamiltonians JF - Journal of Mathematical Physics Y1 - 2018 A1 - Alessandro Michelangeli A1 - Andrea Ottolini A1 - Raffaele Scandone AB -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.

VL - 59 UR - https://doi.org/10.1063/1.5033856 ER - TY - JOUR T1 - Singular Hartree equation in fractional perturbed Sobolev spaces JF - Journal of Nonlinear Mathematical Physics Y1 - 2018 A1 - Alessandro Michelangeli A1 - Alessandro Olgiati A1 - Raffaele Scandone AB -We 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.

PB - Taylor & Francis VL - 25 UR - https://doi.org/10.1080/14029251.2018.1503423 ER - TY - RPRT T1 - On some rigorous aspects of fragmented condensation Y1 - 2018 A1 - Daniele Dimonte A1 - Marco Falconi A1 - Alessandro Olgiati UR - https://arxiv.org/abs/1809.03586 ER - TY - CHAP T1 - Effective Non-linear Dynamics of Binary Condensates and Open Problems T2 - Advances in Quantum Mechanics: Contemporary Trends and Open Problems Y1 - 2017 A1 - Alessandro Olgiati ED - Alessandro Michelangeli ED - Gianfausto Dell'Antonio AB -We 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.

JF - Advances in Quantum Mechanics: Contemporary Trends and Open Problems PB - Springer International Publishing CY - Cham SN - 978-3-319-58904-6 UR - https://doi.org/10.1007/978-3-319-58904-6_14 ER - TY - JOUR T1 - Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates JF - Journal of Nonlinear Mathematical Physics Y1 - 2017 A1 - Alessandro Michelangeli A1 - Alessandro Olgiati AB -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.

PB - Taylor & Francis VL - 24 UR - https://doi.org/10.1080/14029251.2017.1346348 ER - TY - RPRT T1 - Krein-Visik-Birman self-adjoint extension theory revisited Y1 - 2017 A1 - Matteo Gallone A1 - Alessandro Michelangeli A1 - Andrea Ottolini AB - 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. UR - http://preprints.sissa.it/handle/1963/35286 U1 - 35591 U2 - Mathematics ER - TY - JOUR T1 - Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation JF - Advances in Calculus of Variations Y1 - 2017 A1 - Gianni Dal Maso A1 - Gianluca Orlando A1 - Rodica Toader AB -We study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to $L^1$ convergence.

PB - De Gruyter VL - 10 ER - TY - JOUR T1 - Mean-field quantum dynamics for a mixture of Bose–Einstein condensates JF - Analysis and Mathematical Physics Y1 - 2017 A1 - Alessandro Michelangeli A1 - Alessandro Olgiati AB -We study the effective time evolution of a large quantum system consisting of a mixture of different species of identical bosons in interaction. If the system is initially prepared so as to exhibit condensation in each component, we prove that condensation persists at later times and we show quantitatively that the many-body Schrödinger dynamics is effectively described by a system of coupled cubic non-linear Schrödinger equations, one for each component.

VL - 7 UR - https://doi.org/10.1007/s13324-016-0147-3 ER - TY - CHAP T1 - Reduced-order semi-implicit schemes for fluid-structure interaction problems T2 - Model Reduction of Parametrized Systems Y1 - 2017 A1 - Francesco Ballarin A1 - Gianluigi Rozza A1 - Yvon Maday ED - Peter Benner ED - Mario Ohlberger ED - Anthony Patera ED - Gianluigi Rozza ED - Karsten Urban AB -POD–Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.

JF - Model Reduction of Parametrized Systems PB - Springer International Publishing ER - TY - CHAP T1 - Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian T2 - Advances in Quantum Mechanics: Contemporary Trends and Open Problems Y1 - 2017 A1 - Alessandro Olgiati ED - Alessandro Michelangeli ED - Gianfausto Dell'Antonio AB -The 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.

JF - Advances in Quantum Mechanics: Contemporary Trends and Open Problems PB - Springer International Publishing CY - Cham SN - 978-3-319-58904-6 UR - https://doi.org/10.1007/978-3-319-58904-6_15 ER - TY - JOUR T1 - Spectral Properties of the 2+1 Fermionic Trimer with Contact Interactions Y1 - 2017 A1 - Simon Becker A1 - Alessandro Michelangeli A1 - Andrea Ottolini AB - 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. PB - SISSA UR - http://preprints.sissa.it/handle/1963/35303 N1 - Partially supported by the 2014-2017 MIUR-FIR grant \Cond-Math: Condensed Matter and Mathematical Physics" code RBFR13WAET (S.B., A.M., A.O.), by the DAAD International Trainership Programme (S.B.), and by a 2017 visiting research fellowship at the International Center for Mathematical Research CIRM, Trento (A.M.). U1 - 35609 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case JF - Calculus of Variations and Partial Differential Equations Y1 - 2016 A1 - Gianni Dal Maso A1 - Gianluca Orlando A1 - Rodica Toader AB -We study the asymptotic behavior of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter $\varepsilon$, which forces damage concentration on regions of codimension one. We determine the $\Gamma$-limit as $\varepsilon$ tends to zero and show that it contains an energy term involving the crack opening.

VL - 55 UR - https://doi.org/10.1007/s00526-016-0981-z ER - TY - RPRT T1 - Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan--Skornyakov type Y1 - 2016 A1 - Alessandro Michelangeli A1 - Andrea Ottolini AB - 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. UR - http://urania.sissa.it/xmlui/handle/1963/35267 U1 - 35573 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians Y1 - 2016 A1 - Alessandro Michelangeli A1 - Andrea Ottolini AB - 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. UR - http://urania.sissa.it/xmlui/handle/1963/35195 U1 - 35489 U2 - Mathematics U4 - 1 U5 - MAT/07 ER - TY - RPRT T1 - Second-order structured deformations Y1 - 2016 A1 - Ana Cristina Barroso A1 - Jose Matias A1 - Marco Morandotti A1 - David R. Owen PB - SISSA U1 - 35497 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Explicit formulas for relaxed disarrangement densities arising from structured deformations Y1 - 2015 A1 - Ana Cristina Barroso A1 - Jose Matias A1 - Marco Morandotti A1 - David R. Owen AB - Structured deformations provide a multiscale geometry that captures the contributions at the macrolevel of both smooth geometrical changes and non-smooth geometrical changes (disarrangements) at submacroscopic levels. For each (first-order) structured deformation (g,G) of a continuous body, the tensor field G is known to be a measure of deformations without disarrangements, and M:=∇g−G is known to be a measure of deformations due to disarrangements. The tensor fields G and M together deliver not only standard notions of plastic deformation, but M and its curl deliver the Burgers vector field associated with closed curves in the body and the dislocation density field used in describing geometrical changes in bodies with defects. Recently, Owen and Paroni [13] evaluated explicitly some relaxed energy densities arising in Choksi and Fonseca’s energetics of structured deformations [4] and thereby showed: (1) (trM)+ , the positive part of trM, is a volume density of disarrangements due to submacroscopic separations, (2) (trM)−, the negative part of trM, is a volume density of disarrangements due to submacroscopic switches and interpenetrations, and (3) trM, the absolute value of trM, is a volume density of all three of these non-tangential disarrangements: separations, switches, and interpenetrations. The main contribution of the present research is to show that a different approach to the energetics of structured deformations, that due to Ba\'{i}a, Matias, and Santos [1], confirms the roles of (trM)+, (trM)−, and trM established by Owen and Paroni. In doing so, we give an alternative, shorter proof of Owen and Paroni’s results, and we establish additional explicit formulas for other measures of disarrangements. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/34492 U1 - 34687 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Model order reduction of parameterized systems (MoRePaS): Preface to the special issue of advances in computational mathematics JF - Advances in Computational Mathematics Y1 - 2015 A1 - Peter Benner A1 - Mario Ohlberger A1 - Anthony Patera A1 - Gianluigi Rozza A1 - Sorensen, D.C. A1 - Karsten Urban VL - 41 ER - TY - JOUR T1 - Approximate Hitchin-Kobayashi correspondence for Higgs G-bundles Y1 - 2014 A1 - Ugo Bruzzo A1 - Beatriz Graña Otero AB - We announce a result about the extension of the Hitchin-Kobayashi correspondence to principal Higgs bundles. A principal Higgs bundle on a compact Kähler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure. PB - World Scientific Publishing UR - http://urania.sissa.it/xmlui/handle/1963/35095 U1 - 35350 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length Y1 - 2014 A1 - Gianni Dal Maso A1 - Gianluca Orlando A1 - Rodica Toader KW - cracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions AB -We consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.

PB - SISSA UR - http://hdl.handle.net/1963/7271 U1 - 7316 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system JF - Int. Math. Res. Not. (2010) 2010:279-296 Y1 - 2010 A1 - Claudio Bartocci A1 - Gregorio Falqui A1 - Igor Mencattini A1 - Giovanni Ortenzi A1 - Marco Pedroni AB - We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n,R). The relation with the Lax formalism is also discussed. PB - Oxford University Press UR - http://hdl.handle.net/1963/3800 U1 - 8 U2 - LISNU U3 - Interdisciplinary Laboratory for Advanced Studies ER - TY - RPRT T1 - A new model for contact angle hysteresis Y1 - 2007 A1 - Antonio DeSimone A1 - Natalie Gruenewald A1 - Felix Otto AB - We present a model which explains several experimental observations relating contact angle hysteresis with surface roughness. The model is based on the balance between released energy and dissipation, and it describes the stick-slip behavior of drops on a rough surface using ideas similar to those employed in dry friction, elasto-plasticity and fracture mechanics. The main results of our analysis are formulas giving the interval of stable contact angles as a function of the surface roughness. These formulas show that the difference between advancing and receding angles is much larger for a drop in complete contact with the substrate (Wenzel drop) than for one whose cavities are filled with air (Cassie-Baxter drop). This fact is used as the key tool to interpret the experimental evidence. JF - Netw. Heterog. Media 2 (2007) 211-225 UR - http://hdl.handle.net/1963/1848 U1 - 2369 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - 2-d stability of the Néel wall JF - Calc. Var. Partial Differential Equations 27 (2006) 233-253 Y1 - 2006 A1 - Antonio DeSimone A1 - Hans Knuepfer A1 - Felix Otto AB - We are interested in thin-film samples in micromagnetism, where the magnetization m is a 2-d unit-length vector field. More precisely we are interested in transition layers which connect two opposite magnetizations, so called Néel walls. UR - http://hdl.handle.net/1963/2194 U1 - 2050 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - CHAP T1 - Recent analytical developments in micromagnetics T2 - The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381. Y1 - 2006 A1 - Antonio DeSimone A1 - Robert V. Kohn A1 - Stefan Müller A1 - Felix Otto JF - The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381. SN - 978-0-12-480874-4 UR - http://hdl.handle.net/1963/2230 U1 - 2014 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient JF - Nonlinear Analysis, Theory, Methods and Applications. Volume 37, Issue 6, September 1999, Pages 707-717 Y1 - 1999 A1 - Gianni Dal Maso A1 - Vladimir V. Goncharov A1 - Antonio Ornelas AB - A constructive and improved version of the proof that there exist a continuous map that solves the convexified problem is presented. A Lipschitz continuous map is analyzed such that a map vector minimizes the functional at each vector satisfying Cellina\\\'s condition of existence of minimum. This map is explicitly given by a direct constructive algorithm. PB - SISSA UR - http://hdl.handle.net/1963/6439 U1 - 6379 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Renormalized solutions of elliptic equations with general measure data JF - Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 4, 741-808 Y1 - 1999 A1 - Gianni Dal Maso A1 - Francois Murat A1 - Luigi Orsina A1 - Alain Prignet PB - Scuola Normale Superiore di Pisa UR - http://hdl.handle.net/1963/1236 U1 - 2707 U2 - Mathematics U3 - Functional Analysis and Applications ER -