47 research outputs found
Bellman function for extremal problems in BMO
In this paper we develop the method of finding sharp estimates by using a
Bellman function. In such a form the method appears in the proofs of the
classical John--Nirenberg inequality and estimations of BMO functions. In
the present paper we elaborate a method of solving the boundary value problem
for the homogeneous Monge--Amp\`ere equation in a parabolic strip for
sufficiently smooth boundary conditions. In such a way we have obtained an
algorithm of constructing an exact Bellman function for a large class of
integral functionals in the BMO space.Comment: 91 pages, 18 figure
Limitations of the current-phase relation measurements by an asymmetric dc-SQUID
Exotic quantum transport phenomena established in Josephson junctions (JJs)
are reflected by a non-sinusoidal current-phase relation (CPR). The solidified
approach to measure the CPR is via an asymmetric dc-SQUID with a reference JJ
that has a high critical current. We probed this method by measuring CPRs of
hybrid JJs based on a 3D topological insulator (TI) BiTeSe with a
nanobridge acting as a reference JJ. We captured both highly skewed and
sinusoidal critical current oscillations within single devices which
contradicts the uniqueness of the CPR. This implies that the widely used method
provides inaccurate CPR measurement and leads to misinterpretation. It was
shown that the accuracy of the CPR measurement is mediated by the asymmetry in
derivatives of the CPRs but not in critical currents as was previously thought.
We provided considerations for an accurate CPR measurement that encourage
future experiments with reference CPRs different from those that were used
before
Sharp estimates of integral functionals on classes of functions with small mean oscillation
We unify several Bellman function problems into one setting. For that purpose
we define a class of functions that have, in a sense, small mean oscillation
(this class depends on two convex sets in ). We show how the unit
ball in the space, or a Muckenhoupt class, or a Gehring class
can be described in such a fashion. Finally, we consider a Bellman function
problem on these classes, discuss its solution and related questions.Comment: 7 page
Long range coherent magnetic bound states in superconductors
The quantum coherent coupling of completely different degrees of freedom is a
challenging path towards creating new functionalities for quantum electronics.
Usually the antagonistic coupling between spins of magnetic impurities and
superconductivity leads to the destruction of the superconducting order. Here
we show that a localized classical spin of an iron atom immersed in a
superconducting condensate can give rise to new kind of long range coherent
magnetic quantum state. In addition to the well-known Shiba bound state present
on top of an impurity we reveal the existence of a star shaped pattern which
extends as far as 12 nm from the impurity location. This large spatial
dispersion turns out to be related, in a non-trivial way, to the
superconducting coherence length. Inside star branches we observed short scale
interference fringes with a particle-hole asymmetry. Our theoretical approach
captures these features and relates them to the electronic band structure and
the Fermi wave length of the superconductor. The discovery of a directional
long range effect implies that distant magnetic atoms could coherently interact
leading to new topological superconducting phases with fascinating properties