We study the stability problem for a non-relativistic quantum system in dimension three composed by N >= 2 identical fermions, with unit mass, interacting with a different particle, with mass m, via a zero-range interaction of strength alpha is an element of R. We construct the corresponding renormalized quadratic (or energy) form F-alpha and the so-called Skornyakov-Ter-Martirosyan symmetric extension H-alpha, which is the natural candidate as Hamiltonian of the system. We find a value of the mass m*(N) such that for m > m*(N) the form F-alpha is closed and bounded from below. As a consequence, F-alpha 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 F-alpha 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

We study the time evolution in system of N bosons with a relativistic dispersion law interacting through a Newtonian gravitational potential with coupling constant G. We consider the mean field scaling where N tends to infinity, G tends to zero and λ = GN remains fixed. We investigate the relation between the many body quantum dynamics governed by the Schrödinger equation and the effective evolution described by a (semi-relativistic) Hartree equation. In particular, we are interested in the super-critical regime of large λ (the sub-critical case has been studied in [2,19]), where the nonlinear Hartree equation is known to have solutions which blow up in finite time. To inspect this regime, we need to regularize the interaction in the many body Hamiltonian with an N dependent cutoff that vanishes in the limit N → ∞. We show, first, that if the solution of the nonlinear equation does not blow up in the time interval [−T, T ], then the many body Schrödinger dynamics (on the level of the reduced density matrices) can be approximated by the nonlinear Hartree dynamics, just as in the sub-critical regime. Moreover, we prove that if the solution of the nonlinear Hartree equation blows up at time T (in the sense that the H 1/2 norm of the solution diverges as time approaches T ), then also the solution of the linear Schrödinger equation collapses (in the sense that the kinetic energy per particle diverges) if t → T and, simultaneously, N → ∞ sufficiently fast. This gives the first dynamical description of the phenomenon of gravitational collapse as observed directly on the many body level.

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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 Kreȋn, Višik, and Birman. We show that the Ter-Martirosyan-Skornyakov asymptotics is a condition of selfadjointness 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.

We prove that, for arbitrary centres and strengths, the wave operators for three dimensional Schrödinger operators with multi-centre local point interactions are bounded in L p (R 3 ) for 1 < p < 3 and unbounded otherwise.