86 research outputs found
Boundary Behavior of the Ginzburg-Landau Order Parameter in the Surface Superconductivity Regime
We study the 2D Ginzburg-Landau theory for a type-II superconductor in an
applied magnetic field varying between the second and third critical value. In
this regime the order parameter minimizing the GL energy is concentrated along
the boundary of the sample and is well approximated to leading order by a
simplified 1D profile in the direction perpendicular to the boundary. Motivated
by a conjecture of Xing-Bin Pan, we address the question of whether this
approximation can hold uniformly in the boundary region. We prove that this is
indeed the case as a corollary of a refined, second order energy expansion
including contributions due to the curvature of the sample. Local variations of
the GL order parameter are controlled by the second order term of this energy
expansion, which allows us to prove the desired uniformity of the surface
superconductivity layer
Quantum Mechanics and Stochastic Mechanics for compatible observables at different times
Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum
Mechanics in presence of non-interacting subsystems. In both cases, it is shown
that correlations at different times of compatible position observables on
stationary states agree with Quantum Mechanics only in the case of product wave
functions. By appropriate Bell-like inequalities it is shown that no classical
theory, in particular no stochastic process, can reproduce the quantum
mechanical correlations of position variables of non interacting systems at
different times.Comment: Plain Te
Magnetic Schr\"odinger Operators as the Quasi-Classical Limit of Pauli-Fierz-type Models
We study the quasi-classical limit of the Pauli-Fierz model: the system is
composed of finitely many non-relativistic charged particles interacting with a
bosonic radiation field. We trace out the degrees of freedom of the field, and
consider the classical limit of the latter. We prove that the partial trace of
the full Hamiltonian converges, in resolvent sense, to an effective
Schr\"odinger operator with magnetic field and a corrective electric potential
that depends on the field configuration. Furthermore, we prove the convergence
of the ground state energy of the microscopic system to the infimum over all
possible classical field configurations of the ground state energy of the
effective Schr\"odinger operator.Comment: 26 pages, pdfLatex. Final version to appear in J. Spectr. Theor
Surface Superconductivity in Presence of Corners
We consider an extreme type-II superconducting wire with non-smooth cross
section, i.e., with one or more corners at the boundary, in the framework of
the Ginzburg-Landau theory. We prove the existence of an interval of values of
the applied field, where superconductivity is spread uniformly along the
boundary of the sample. More precisely the energy is not affected to leading
order by the presence of corners and the modulus of the Ginzburg-Landau
minimizer is approximately constant along the transversal direction. The
critical fields delimiting this surface superconductivity regime coincide with
the ones in absence of boundary singularities.Comment: 20 pages, pdfLaTex, 2 figure
Local Density Approximation for Almost-Bosonic Anyons
We discuss the average-field approximation for a trapped gas of
non-interacting anyons in the quasi-bosonic regime. In the homogeneous case,
i.e., for a confinement to a bounded region, we prove that the energy in the
regime of large statistics parameter, i.e., for "less-bosonic" anyons, is
independent of boundary conditions and of the shape of the domain. When a
non-trivial trapping potential is present, we derive a local density
approximation in terms of a Thomas-Fermi-like model.Comment: Contribution to the proceedings of QMath13: Mathematical Results in
Quantum Physics, 8-11 October 2016, Atlanta, U
Two-dimensional Time-dependent Point Interactions
We study the time-evolution of a quantum particle subjected to time-dependent
zero-range forces in two dimensions. After establishing a conceivable ansatz
for the solution to the Schr\"{o}dinger equation, we prove that the wave packet
time-evolution is completely specified by the solutions of a system of
Volterra-type equations -- the {\it charge equations} -- involving the
coefficients of the singular part of the wave function, thus extending to the
two-dimensional case known results in one and three dimensions.Comment: 17 pages, AMS-LaTex; presentation of the model changed, small changes
to Lemma 2.1 and Proposition 2.
Rapidly Rotating Bose-Einstein Condensates in Homogeneous Traps
We extend the results of a previous paper on the Gross-Pitaevskii description
of rotating Bose-Einstein condensates in two-dimensional traps to confining
potentials of the form V(r) = r^s, . Writing the coupling constant
as we study the limit . We derive rigorously the
leading asymptotics of the ground state energy and the density profile when the
rotation velocity \Omega tends to infinity as a power of . The case
of asymptotically homogeneous potentials is also discussed.Comment: LaTex2e, 16 page
Vortex Phases of Rotating Superfluids
We report on the first mathematically rigorous proofs of a transition to a
giant vortex state of a superfluid in rotating anharmonic traps. The analysis
is carried out within two-dimensional Gross-Pitaevskii theory at large coupling
constant and large rotational velocity and is based on precise asymptotic
estimates on the ground state energy. An interesting aspect is a significant
difference between 'soft' anharmonic traps (like a quartic plus quadratic
trapping potential) and traps with a fixed boundary. In the former case
vortices persist in the bulk until the width of the annulus becomes comparable
to the size of the vortex cores. In the second case the transition already
takes place in a parameter regime where the size of vortices is very small
relative to the width of the annulus. Moreover, the density profiles in the
annulus are different in the two cases. In both cases rotational symmetry of
the density in a true ground state is broken, even though a symmetric
variational ansatz gives an excellent approximation to the energy.Comment: For the Proceedings of 21st International Laser Physics Workshop,
Calgary, July 23-27, 201
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