198 research outputs found
Nucleation of bulk superconductivity close to critical magnetic field
We consider the two-dimensional Ginzburg-Landau functional with constant
applied magnetic field. For applied magnetic fields close to the second
critical field and large Ginzburg-Landau parameter, we provide
leading order estimates on the energy of minimizing configurations. We obtain a
fine threshold value of the applied magnetic field for which bulk
superconductivity contributes to the leading order of the energy. Furthermore,
the energy of the bulk is related to that of the Abrikosov problem in a
periodic lattice. A key ingredient of the proof is a novel -bound
which is of independent interest
On the energy of bound states for magnetic Schr\"odinger operators
We provide a leading order semiclassical asymptotics of the energy of bound
states for magnetic Neumann Schr\"odinger operators in two dimensional
(exterior) domains with smooth boundaries. The asymptotics is valid all the way
up to the bottom of the essential spectrum. When the spectral parameter is
varied near the value where bound states become allowed in the interior of the
domain, we show that the energy has a boundary and a bulk component. The
estimates rely on coherent states, in particular on the construction of
`boundary coherent states', and magnetic Lieb-Thirring estimates.Comment: 26 page
Lack of diamagnetism and the Little-Parks effect
When a superconducting sample is submitted to a sufficiently strong external
magnetic field, the superconductivity of the material is lost. In this paper we
prove that this effect does not, in general, take place at a unique value of
the external magnetic field strength. Indeed, for a sample in the shape of a
narrow annulus the set of magnetic field strengths for which the sample is
superconducting is not an interval. This is a rigorous justification of the
Little-Parks effect. We also show that the same oscillation effect can happen
for disc-shaped samples if the external magnetic field is non-uniform. In this
case the oscillations can even occur repeatedly along arbitrarily large values
of the Ginzburg--Landau parameter . The analysis is based on an
understanding of the underlying spectral theory for a magnetic Schr\"{o}dinger
operator. It is shown that the ground state energy of such an operator is not
in general a monotone function of the intensity of the field, even in the limit
of strong fields
The electron densities of pseudorelativistic eigenfunctions are smooth away from the nuclei
We consider a pseudorelativistic model of atoms and molecules, where the
kinetic energy of the electrons is given by . In this model
the eigenfunctions are generally not even bounded, however, we prove that the
corresponding one-electron densities are smooth away from the nuclei.Comment: 16 page
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