70 research outputs found
The singular Hartree equation in fractional perturbed Sobolev spaces
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\"odinger 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.Comment: Published on Journal of Nonlinear Mathematical Physics (2018
Discrete spectra for critical Dirac-Coulomb Hamiltonians
The one-particle Dirac Hamiltonian with Coulomb interaction is known to be
realised, in a regime of large (critical) couplings, by an infinite
multiplicity of distinct self-adjoint operators, including a distinguished,
physically most natural one. For the latter, Sommerfeld's celebrated fine
structure formula provides the well-known expression for the eigenvalues in the
gap of the continuum spectrum. Exploiting our recent general classification of
all other self-adjoint realisations, we generalise Sommerfeld's formula so as
to determine the discrete spectrum of all other self-adjoint versions of the
Dirac-Coulomb Hamiltonian. Such discrete spectra display naturally a fibred
structure, whose bundle covers the whole gap of the continuum spectrum.Comment: 24 pages, 3 figures. Version published on Journal of Mathematical
Physics (2018
Point-like perturbed fractional Laplacians through shrinking potentials of finite range
We reconstruct the rank-one, singular (point-like) perturbations of the
-dimensional fractional Laplacian in the physically meaningful
norm-resolvent limit of fractional Schr\"{o}dinger operators with regular
potentials centred around the perturbation point and shrinking to a delta-like
shape. We analyse both the possible regimes, the resonance-driven and the
resonance-independent limit, depending on the power of the fractional Laplacian
and the spatial dimension. To this aim, we also qualify the notion of
zero-energy resonance for Schr\"{o}dinger operators formed by a fractional
Laplacian and a regular potential
Fractional powers and singular perturbations of quantum differential Hamiltonians
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 such two 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.Comment: Published on J. Math. Phys. (2018
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