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{\textendash}Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross{\textendash}Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is O(1), we are able to verify Bogoliubov{\textquoteright}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.

}, doi = {10.1142/S0129055X19500053}, url = {https://doi.org/10.1142/S0129055X19500053}, author = {Alessandro Michelangeli and Phan Thanh Nam and Alessandro Olgiati} } @article {doi:10.1142/S0218202518500379, title = {Cohesive fracture with irreversibility: Quasistatic evolution for a model subject to fatigue}, journal = {Mathematical Models and Methods in Applied Sciences}, volume = {28}, number = {07}, year = {2018}, pages = {1371-1412}, abstract = {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.

}, doi = {10.1142/S0218202518500379}, url = {https://doi.org/10.1142/S0218202518500379}, author = {Vito Crismale and Giuliano Lazzaroni and Gianluca Orlando} } @article {Michelangeli_2018, title = {Effective non-linear spinor dynamics in a spin-1 Bose{\textendash}Einstein condensate}, journal = {Journal of Physics A: Mathematical and Theoretical}, volume = {51}, number = {40}, year = {2018}, month = {sep}, pages = {405201}, publisher = {IOP Publishing}, abstract = {We derive from first principles the experimentally observed effective dynamics of a spinor Bose gas initially prepared as a Bose{\textendash}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{\textendash}spin interaction. The effective dynamics is well-known to be governed by a system of coupled semi-linear Schr{\"o}dinger 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.

}, doi = {10.1088/1751-8121/aadbc2}, url = {https://doi.org/10.1088\%2F1751-8121\%2Faadbc2}, author = {Alessandro Michelangeli and Alessandro Olgiati} } @article {doi:10.1063/1.5033856, title = {Fractional powers and singular perturbations of quantum differential Hamiltonians}, journal = {Journal of Mathematical Physics}, volume = {59}, number = {7}, year = {2018}, pages = {072106}, abstract = {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 Schr{\"o}dinger equations for the corresponding operators, we outline a programme of relevant questions that deserve being investigated.

}, doi = {10.1063/1.5033856}, url = {https://doi.org/10.1063/1.5033856}, author = {Alessandro Michelangeli and Andrea Ottolini and Raffaele Scandone} } @article {doi:10.1080/14029251.2018.1503423, title = {Singular Hartree equation in fractional perturbed Sobolev spaces}, journal = {Journal of Nonlinear Mathematical Physics}, volume = {25}, number = {4}, year = {2018}, pages = {558-588}, publisher = {Taylor \& Francis}, abstract = {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 Schr{\"o}dinger 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.

}, doi = {10.1080/14029251.2018.1503423}, url = {https://doi.org/10.1080/14029251.2018.1503423}, author = {Alessandro Michelangeli and Alessandro Olgiati and Raffaele Scandone} } @article {1809.03586, title = {On some rigorous aspects of fragmented condensation}, year = {2018}, url = {https://arxiv.org/abs/1809.03586}, author = {Daniele Dimonte and Marco Falconi and Alessandro Olgiati} } @inbook {Olgiati2017, title = {Effective Non-linear Dynamics of Binary Condensates and Open Problems}, booktitle = {Advances in Quantum Mechanics: Contemporary Trends and Open Problems}, year = {2017}, pages = {239{\textendash}256}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {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 Sch{\"o}dinger 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.

}, isbn = {978-3-319-58904-6}, doi = {10.1007/978-3-319-58904-6_14}, url = {https://doi.org/10.1007/978-3-319-58904-6_14}, author = {Alessandro Olgiati}, editor = {Alessandro Michelangeli and Gianfausto Dell{\textquoteright}Antonio} } @article {doi:10.1080/14029251.2017.1346348, title = {Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates}, journal = {Journal of Nonlinear Mathematical Physics}, volume = {24}, number = {3}, year = {2017}, pages = {426-464}, publisher = {Taylor \& Francis}, abstract = {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 Schr{\"o}dinger 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 Schr{\"o}dinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl{\textquoteright}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.

}, doi = {10.1080/14029251.2017.1346348}, url = {https://doi.org/10.1080/14029251.2017.1346348}, author = {Alessandro Michelangeli and Alessandro Olgiati} } @article {2017, title = {Krein-Visik-Birman self-adjoint extension theory revisited}, number = {SISSA;25/2017/MATE}, year = {2017}, abstract = {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.}, url = {http://preprints.sissa.it/handle/1963/35286}, author = {Matteo Gallone and Alessandro Michelangeli and Andrea Ottolini} } @article {dal2017lower, title = {Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation}, journal = {Advances in Calculus of Variations}, volume = {10}, number = {2}, year = {2017}, pages = {183{\textendash}207}, publisher = {De Gruyter}, abstract = {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.

}, doi = {10.1515/acv-2015-0036}, author = {Gianni Dal Maso and Gianluca Orlando and Rodica Toader} } @article {Michelangeli2017, title = {Mean-field quantum dynamics for a mixture of Bose{\textendash}Einstein condensates}, journal = {Analysis and Mathematical Physics}, volume = {7}, number = {4}, year = {2017}, month = {Dec}, pages = {377{\textendash}416}, abstract = {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 Schr{\"o}dinger dynamics is effectively described by a system of coupled cubic non-linear Schr{\"o}dinger equations, one for each component.

}, issn = {1664-235X}, doi = {10.1007/s13324-016-0147-3}, url = {https://doi.org/10.1007/s13324-016-0147-3}, author = {Alessandro Michelangeli and Alessandro Olgiati} } @inbook {BallarinRozzaMaday2017, title = {Reduced-order semi-implicit schemes for fluid-structure interaction problems}, booktitle = {Model Reduction of Parametrized Systems}, year = {2017}, pages = {149{\textendash}167}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, chapter = {Reduced-order semi-implicit schemes for fluid-structure interaction problems}, abstract = {POD{\textendash}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.

}, doi = {10.1007/978-3-319-58786-8_10}, author = {Francesco Ballarin and Gianluigi Rozza and Yvon Maday}, editor = {Peter Benner and Mario Ohlberger and Anthony Patera and Gianluigi Rozza and Karsten Urban} } @inbook {Olgiati2017, title = {Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian}, booktitle = {Advances in Quantum Mechanics: Contemporary Trends and Open Problems}, year = {2017}, pages = {257{\textendash}266}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {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 {\textquoteleft}{\textquoteleft}projection counting{\textquoteright}{\textquoteright} scheme.

}, isbn = {978-3-319-58904-6}, doi = {10.1007/978-3-319-58904-6_15}, url = {https://doi.org/10.1007/978-3-319-58904-6_15}, author = {Alessandro Olgiati}, editor = {Alessandro Michelangeli and Gianfausto Dell{\textquoteright}Antonio} } @article {2017, title = {Spectral Properties of the 2+1 Fermionic Trimer with Contact Interactions}, number = {SISSA;61/2017/MATE}, year = {2017}, note = {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.).}, publisher = {SISSA}, abstract = {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.}, url = {http://preprints.sissa.it/handle/1963/35303}, author = {Simon Becker and Alessandro Michelangeli and Andrea Ottolini} } @article {DalMaso2016, title = {Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {55}, number = {3}, year = {2016}, month = {Apr}, pages = {45}, abstract = {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.

}, issn = {1432-0835}, doi = {10.1007/s00526-016-0981-z}, url = {https://doi.org/10.1007/s00526-016-0981-z}, author = {Gianni Dal Maso and Gianluca Orlando and Rodica Toader} } @article {2016, title = {Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan--Skornyakov type}, number = {SISSA;65/2016/MATE}, year = {2016}, abstract = {We reconstruct the whole family of self-adjoint Hamiltonians of Ter-Martirosyan- Skornyakov type for a system of two identical fermions coupled with a third particle of different nature through an interaction of zero range. We proceed through an operator-theoretic approach based on the self-adjoint extension theory of Krein, Vi{\v s}iik, and Birman. We identify the explicit {\textquoteleft}Krein-Vi{\v s}ik-Birman extension param- eter{\textquoteright} as an operator on the {\textquoteleft}space of charges{\textquoteright} for this model (the {\textquoteleft}Krein space{\textquoteright}) 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.}, url = {http://urania.sissa.it/xmlui/handle/1963/35267}, author = {Alessandro Michelangeli and Andrea Ottolini} } @article {2016, title = {On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians}, number = {SISSA;11/2016/MATE}, year = {2016}, abstract = {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 {\textquoteleft}{\textquoteleft}Ter-Martirosyan--Skornyakov condition{\textquoteright}{\textquoteright} 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.}, url = {http://urania.sissa.it/xmlui/handle/1963/35195}, author = {Alessandro Michelangeli and Andrea Ottolini} } @article {2016, title = {Second-order structured deformations}, year = {2016}, institution = {SISSA}, author = {Ana Cristina Barroso and Jose Matias and Marco Morandotti and David R. Owen} } @article {2015, title = {Explicit formulas for relaxed disarrangement densities arising from structured deformations}, number = {SISSA;37/2015/MATE}, year = {2015}, institution = {SISSA}, abstract = {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{\textquoteright}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{\textquoteright}s results, and we establish additional explicit formulas for other measures of disarrangements.}, url = {http://urania.sissa.it/xmlui/handle/1963/34492}, author = {Ana Cristina Barroso and Jose Matias and Marco Morandotti and David R. Owen} } @article {BennerOhlbergerPateraRozzaSorensenUrban2015, title = {Model order reduction of parameterized systems ({MoRePaS}): Preface to the special issue of advances in computational mathematics}, journal = {Advances in Computational Mathematics}, volume = {41}, number = {5}, year = {2015}, pages = {955{\textendash}960}, doi = {10.1007/s10444-015-9443-y}, author = {Peter Benner and Mario Ohlberger and Anthony Patera and Gianluigi Rozza and Sorensen, D.C. and Karsten Urban} } @article {2014, title = {Approximate Hitchin-Kobayashi correspondence for Higgs G-bundles}, number = {International Journal of Geometric Methods in Modern Physics;volume 11; issue 7; article number 1460015;}, year = {2014}, publisher = {World Scientific Publishing}, abstract = {We announce a result about the extension of the Hitchin-Kobayashi correspondence to principal Higgs bundles. A principal Higgs bundle on a compact K{\"a}hler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.}, doi = {10.1142/S0219887814600159}, url = {http://urania.sissa.it/xmlui/handle/1963/35095}, author = {Ugo Bruzzo and Beatriz Gra{\~n}a Otero} } @article {2014, title = {Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length}, number = {Nonlinear Differential Equations and Applications}, year = {2014}, publisher = {SISSA}, abstract = {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.

}, keywords = {cracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions}, doi = {10.1007/s00030-014-0291-0}, url = {http://hdl.handle.net/1963/7271}, author = {Gianni Dal Maso and Gianluca Orlando and Rodica Toader} } @article {2010, title = {On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system}, journal = {Int. Math. Res. Not. (2010) 2010:279-296}, number = {arXiv.org;0902.0953v2}, year = {2010}, publisher = {Oxford University Press}, abstract = {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.}, doi = {10.1093/imrn/rnp130}, url = {http://hdl.handle.net/1963/3800}, author = {Claudio Bartocci and Gregorio Falqui and Igor Mencattini and Giovanni Ortenzi and Marco Pedroni} } @article {2007, title = {A new model for contact angle hysteresis}, number = {SISSA;37/2006/M}, year = {2007}, abstract = {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.}, url = {http://hdl.handle.net/1963/1848}, author = {Antonio DeSimone and Natalie Gruenewald and Felix Otto} } @article {2006, title = {2-d stability of the N{\'e}el wall}, journal = {Calc. Var. Partial Differential Equations 27 (2006) 233-253}, year = {2006}, abstract = {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 N{\'e}el walls.}, doi = {10.1007/s00526-006-0019-z}, url = {http://hdl.handle.net/1963/2194}, author = {Antonio DeSimone and Hans Knuepfer and Felix Otto} } @inbook {2006, title = {Recent analytical developments in micromagnetics}, booktitle = {The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381.}, number = {SISSA;76/2004/M}, year = {2006}, isbn = {978-0-12-480874-4}, url = {http://hdl.handle.net/1963/2230}, author = {Antonio DeSimone and Robert V. Kohn and Stefan M{\"u}ller and Felix Otto} } @article {1999, title = {A Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient}, journal = {Nonlinear Analysis, Theory, Methods and Applications. Volume 37, Issue 6, September 1999, Pages 707-717}, year = {1999}, publisher = {SISSA}, abstract = {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.}, doi = {10.1016/S0362-546X(98)00067-4}, url = {http://hdl.handle.net/1963/6439}, author = {Gianni Dal Maso and Vladimir V. Goncharov and Antonio Ornelas} } @article {1999, title = {Renormalized solutions of elliptic equations with general measure data}, journal = {Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 4, 741-808}, number = {SISSA;22/99/M}, year = {1999}, publisher = {Scuola Normale Superiore di Pisa}, url = {http://hdl.handle.net/1963/1236}, author = {Gianni Dal Maso and Francois Murat and Luigi Orsina and Alain Prignet} }