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.

1 aMichelangeli, Alessandro1 aOttolini, Andrea1 aScandone, Raffaele uhttps://doi.org/10.1063/1.503385600742nas 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/1963/3528601253nas 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/1963/3526701116nas 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/1963/35195