- Derivation and study of effective non-linear Schrödinger
equations from quantum many-body dynamics.
- Effective dynamics of many-body boson and fermion systems in
the mean-field limit of infinitely many particles.
- Short-range, hard-core scaling limits (the Gross-Pitaevski
- Effective dynamics under small and large external magnetic
- Linear and non-linear PDEs for the dynamics of quantum
systems: well-posedness and scattering.
- Non-linear Schrödinger and Hartree equations:
non-relativistic, relativistic, magnetic.
- Schrödinger-Maxwell and Hartree-Maxwell systems.
- Magnetic Strichartz estimates.
- Few-body and many-body quantum systems with zero-range
- Singular perturbations of elliptic operators by means of the
theory of self-adjoint extension of symmetric operators on a Hilbert
space and the theory of quadratic forms on a Hilbert space.
- Scaling limits on Schrödinger Hamiltonians to obtain
- Spectral properties of point-interaction Hamiltonians:
collapse of the 3-body system (the Thomas effect), bound states,
relaxation at zero-energy (the Efimov effect).
- Role of zero-energy resonances for finite-range potentials
- Point-interactions in finite domains and delta-like
interactions on surfaces.
- The mathematics of quantum particles constrained on graphs.
- Quantum models on graphs as the limiting description for
nanotubes and complex molecules.
- Schrödinger dynamics on waveguides with Dirichlet or Neumann
- Effective dynamics in the limit of a waveguide shrinking to a
graph. Emergence of boundary conditions at the graph vertices.
- Non-linear Schrödinger equations on graphs and soliton
- Mathematical models for quantum decoherence.
- Quantum models of decoherence induced by the environment.
- The measure process in Quantum Mechanics.
- Emergence of straight tracks in a tracking chamber with no
reference to the wave packet collapse postulate (the Mott's problem).
Time-dependent perturbative analysis.
- Equilibrium and non-equilibrium in Quantum Statistical
Mechanics with infinitely many degrees of freedom.
- C*-algebraic formulation of Quantum Mechanics and Quantum
- Von-Neumann algebras and Tomita-Takesaki theory.
- Steady states vs Equilibrium (KMS) states.