48 research outputs found
Surface Effects in Superconductors with Corners
We review some recent results on the phenomenon of surface superconductivity
in the framework of Ginzburg-Landau theory for extreme type-II materials. In
particular, we focus on the response of the superconductor to a strong
longitudinal magnetic field in the regime where superconductivity survives only
along the boundary of the wire. We derive the energy and density asymptotics
for samples with smooth cross section, up to curvature-dependent terms.
Furthermore, we discuss the corrections in presence of corners at the boundary
of the sample.Comment: Proceeding for XXI Congresso UMI 2019, final version, Boll. Unione
Mat. Ital. to appear, 17 pages, pdfLaTe
Energy lower bound for the unitary N+1 fermionic model
We consider the stability problem for a unitary N+1 fermionic model, i.e., a
system of identical fermions interacting via zero-range interactions with a
different particle, in the case of infinite two-body scattering length. We
present a slightly more direct and simplified proof of a recent result obtained
in \cite{CDFMT}, where a sufficient stability condition is proved under a
suitable assumption on the mass ratio.Comment: 7 page
Validity of spin wave theory for the quantum Heisenberg model
Spin wave theory is a key ingredient in our comprehension of quantum spin
systems, and is used successfully for understanding a wide range of magnetic
phenomena, including magnon condensation and stability of patterns in dipolar
systems. Nevertheless, several decades of research failed to establish the
validity of spin wave theory rigorously, even for the simplest models of
quantum spins. A rigorous justification of the method for the three-dimensional
quantum Heisenberg ferromagnet at low temperatures is presented here. We derive
sharp bounds on its free energy by combining a bosonic formulation of the model
introduced by Holstein and Primakoff with probabilistic estimates and operator
inequalities.Comment: 4 page
Well-posedness of the Two-dimensional Nonlinear Schr\"odinger Equation with Concentrated Nonlinearity
We consider a two-dimensional nonlinear Schr\"odinger equation with
concentrated nonlinearity. In both the focusing and defocusing case we prove
local well-posedness, i.e., existence and uniqueness of the solution for short
times, as well as energy and mass conservation. In addition, we prove that this
implies global existence in the defocusing case, irrespective of the power of
the nonlinearity, while in the focusing case blowing-up solutions may arise.Comment: 39 pages, pdfLaTex. Final version to appear in Ann. I. H. Poincar\'e
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Universal and shape dependent features of surface superconductivity
We analyze the response of a type II superconducting wire to an external
magnetic field parallel to it in the framework of Ginzburg-Landau theory. We
focus on the surface superconductivity regime of applied field between the
second and third critical values, where the superconducting state survives only
close to the sample's boundary. Our first finding is that, in first
approximation, the shape of the boundary plays no role in determining the
density of superconducting electrons. A second order term is however isolated,
directly proportional to the mean curvature of the boundary. This demonstrates
that points of higher boundary curvature (counted inwards) attract
superconducting electrons
Vortex patterns in the almost-bosonic anyon gas
We study theoretically and numerically the ground state of a gas of 2D
abelian anyons in an external trapping potential. We treat anyon statistics in
the magnetic gauge picture, perturbatively around the bosonic end. This leads
to a mean-field energy functional, whose ground state displays vortex lattices
similar to those found in rotating Bose-Einstein condensates. A crucial
difference is however that the vortex density is proportional to the underlying
matter density of the gas
Effects of boundary curvature on surface superconductivity
We investigate, within 2D Ginzburg-Landau theory, the ground state of a type-II superconducting cylinder in a parallel magnetic field varying between the second and third critical values. In this regime, superconductivity is restricted to a thin shell along the boundary of the sample and is to leading order constant in the direction tangential to the boundary. We exhibit a correction to this effect, showing that the curvature of the sample affects the distribution of superconductivity
Ionization for Three Dimensional Time-dependent Point Interactions
We study the time evolution of a three dimensional quantum particle under the
action of a time-dependent point interaction fixed at the origin. We assume
that the ``strength'' of the interaction (\alpha(t)) is a periodic function
with an arbitrary mean. Under very weak conditions on the Fourier coefficients
of (\alpha(t)), we prove that there is complete ionization as (t \to \infty),
starting from a bound state at time (t = 0). Moreover we prove also that, under
the same conditions, all the states of the system are scattering states.Comment: Some improvements and some references added, 26 pages, LaTe
Ground State Properties in the Quasi-Classical Regime
We study the ground state energy and ground states of systems coupling
non-relativistic quantum particles and force-carrying Bose fields, such as
radiation, in the quasi-classical approximation. The latter is very useful
whenever the force-carrying field has a very large number of excitations,and
thus behaves in a semiclassical way, while the non-relativistic particles, on
the other hand, retain their microscopic features. We prove that the ground
state energy of the fully microscopic model converges to the one of a nonlinear
quasi-classical functional depending on both the particles' wave function and
the classical configuration of the field. Equivalently, this energy can be
interpreted as the lowest energy of a Pekar-like functional with an effective
nonlinear interaction for the particles only. If the particles are confined,
the ground state of the microscopic system converges as well, to a probability
measure concentrated on the set of minimizers of the quasi-classical energy.Comment: 52 pages, pdfLaTe