Browsing by Author "Ottolini, Andrea"

Now showing 1 - 5 of 5
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    Fractional powers and singular perturbations of quantum differential Hamiltonians
    (SISSA, 2018-01) Michelangeli, Alessandro; Ottolini, Andrea; Scandone, Raffaele; Mathematics
    We consider the fractional powers of singular (point-like) perturbations of the Laplacian and the singular perturbations of fractional powers of the Laplacian, and we compare two such constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schrödinger equations for the corresponding operators, we outline a programme of relevant questions that deserve being investigated.
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    Krein-Visik-Birman self-adjoint extension theory revisited
    (2017) Gallone, Matteo; Michelangeli, Alessandro; Ottolini, Andrea; Mathematics
    The core results of the so-called KreIn-Visik-Birman theory of self-adjoint extensions of semi-bounded symmetric operators are reproduced, both in their original and in a more modern formulation, within a comprehensive discussion that includes missing details, elucidative steps, and intermediate results of independent interest.
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    Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan--Skornyakov type
    (2016-12) Michelangeli, Alessandro; Ottolini, Andrea; Mathematics
    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, Višik, and Birman. We identify the explicit ‘Kreĭn-Višik–Birman extension parameter’ 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 reproduce for the first time the previous partial constructions obtained by means of an alternative quadratic form approach.
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    On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians
    (2016) Michelangeli, Alessandro; Ottolini, Andrea; Mathematics
    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.
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    Spectral Properties of the 2+1 Fermionic Trimer with Contact Interactions
    (SISSA, 2017-12-21) Becker, Simon; Michelangeli, Alessandro; Ottolini, Andrea; Mathematics
    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.