3,056 research outputs found
Stability of the magnetic Schr\"odinger operator in a waveguide
The spectrum of the Schr\"odinger operator in a quantum waveguide is known to
be unstable in two and three dimensions. Any enlargement of the waveguide
produces eigenvalues beneath the continuous spectrum. Also if the waveguide is
bent eigenvalues will arise below the continuous spectrum. In this paper a
magnetic field is added into the system. The spectrum of the magnetic
Schr\"odinger operator is proved to be stable under small local deformations
and also under small bending of the waveguide. The proof includes a magnetic
Hardy-type inequality in the waveguide, which is interesting in its own
Strong-coupling asymptotic expansion for Schr\"odinger operators with a singular interaction supported by a curve in
We investigate a class of generalized Schr\"{o}dinger operators in
with a singular interaction supported by a smooth curve
. We find a strong-coupling asymptotic expansion of the discrete
spectrum in case when is a loop or an infinite bent curve which is
asymptotically straight. It is given in terms of an auxiliary one-dimensional
Schr\"{o}dinger operator with a potential determined by the curvature of
. In the same way we obtain an asymptotics of spectral bands for a
periodic curve. In particular, the spectrum is shown to have open gaps in this
case if is not a straight line and the singular interaction is strong
enough.Comment: LaTeX 2e, 30 pages; minor improvements, to appear in Rev. Math. Phy
Relativistic Scott correction in self-generated magnetic fields
We consider a large neutral molecule with total nuclear charge in a model
with self-generated classical magnetic field and where the kinetic energy of
the electrons is treated relativistically. To ensure stability, we assume that
, where denotes the fine structure constant. We are
interested in the ground state energy in the simultaneous limit , such that is fixed. The
leading term in the energy asymptotics is independent of , it is given
by the Thomas-Fermi energy of order and it is unchanged by including
the self-generated magnetic field. We prove the first correction term to this
energy, the so-called Scott correction of the form . The
current paper extends the result of \cite{SSS} on the Scott correction for
relativistic molecules to include a self-generated magnetic field. Furthermore,
we show that the corresponding Scott correction function , first identified
in \cite{SSS}, is unchanged by including a magnetic field. We also prove new
Lieb-Thirring inequalities for the relativistic kinetic energy with magnetic
fields.Comment: Small typos corrected, new references adde
Abelian subgroups of Garside groups
In this paper, we show that for every abelian subgroup of a Garside
group, some conjugate consists of ultra summit elements and the
centralizer of is a finite index subgroup of the normalizer of .
Combining with the results on translation numbers in Garside groups, we obtain
an easy proof of the algebraic flat torus theorem for Garside groups and solve
several algorithmic problems concerning abelian subgroups of Garside groups.Comment: This article replaces our earlier preprint "Stable super summit sets
in Garside groups", arXiv:math.GT/060258
Scattering through a straight quantum waveguide with combined boundary conditions
Scattering through a straight two-dimensional quantum waveguide Rx(0,d) with
Dirichlet boundary conditions on (-\infty,0)x{y=0} \cup (0,\infty)x{y=d} and
Neumann boundary condition on (-infty,0)x{y=d} \cup (0,\infty)x{y=0} is
considered using stationary scattering theory. The existence of a matching
conditions solution at x=0 is proved. The use of stationary scattering theory
is justified showing its relation to the wave packets motion. As an
illustration, the matching conditions are also solved numerically and the
transition probabilities are shown.Comment: 26 pages, 3 figure
Fisheye Consistency: Keeping Data in Synch in a Georeplicated World
Over the last thirty years, numerous consistency conditions for replicated
data have been proposed and implemented. Popular examples of such conditions
include linearizability (or atomicity), sequential consistency, causal
consistency, and eventual consistency. These consistency conditions are usually
defined independently from the computing entities (nodes) that manipulate the
replicated data; i.e., they do not take into account how computing entities
might be linked to one another, or geographically distributed. To address this
lack, as a first contribution, this paper introduces the notion of proximity
graph between computing nodes. If two nodes are connected in this graph, their
operations must satisfy a strong consistency condition, while the operations
invoked by other nodes are allowed to satisfy a weaker condition. The second
contribution is the use of such a graph to provide a generic approach to the
hybridization of data consistency conditions into the same system. We
illustrate this approach on sequential consistency and causal consistency, and
present a model in which all data operations are causally consistent, while
operations by neighboring processes in the proximity graph are sequentially
consistent. The third contribution of the paper is the design and the proof of
a distributed algorithm based on this proximity graph, which combines
sequential consistency and causal consistency (the resulting condition is
called fisheye consistency). In doing so the paper not only extends the domain
of consistency conditions, but provides a generic provably correct solution of
direct relevance to modern georeplicated systems
Quasi-classical versus non-classical spectral asymptotics for magnetic Schroedinger operators with decreasing electric potentials
We consider the Schroedinger operator H on L^2(R^2) or L^2(R^3) with constant
magnetic field and electric potential V which typically decays at infinity
exponentially fast or has a compact support. We investigate the asymptotic
behaviour of the discrete spectrum of H near the boundary points of its
essential spectrum. If the decay of V is Gaussian or faster, this behaviour is
non-classical in the sense that it is not described by the quasi-classical
formulas known for the case where V admits a power-like decay.Comment: Corrected versio
Topologically protected quantum gates for computation with non-Abelian anyons in the Pfaffian quantum Hall state
We extend the topological quantum computation scheme using the Pfaffian
quantum Hall state, which has been recently proposed by Das Sarma et al., in a
way that might potentially allow for the topologically protected construction
of a universal set of quantum gates. We construct, for the first time, a
topologically protected Controlled-NOT gate which is entirely based on
quasihole braidings of Pfaffian qubits. All single-qubit gates, except for the
pi/8 gate, are also explicitly implemented by quasihole braidings. Instead of
the pi/8 gate we try to construct a topologically protected Toffoli gate, in
terms of the Controlled-phase gate and CNOT or by a braid-group based
Controlled-Controlled-Z precursor. We also give a topologically protected
realization of the Bravyi-Kitaev two-qubit gate g_3.Comment: 6 pages, 7 figures, RevTeX; version 3: introduced section names, new
reference added; new comment added about the embedding of the one- and two-
qubit gates into a three-qubit syste
Sufficient conditions for the existence of bound states in a central potential
We show how a large class of sufficient conditions for the existence of bound
states, in non-positive central potentials, can be constructed. These
sufficient conditions yield upper limits on the critical value,
, of the coupling constant (strength), , of the
potential, , for which a first -wave bound state appears.
These upper limits are significantly more stringent than hitherto known
results.Comment: 7 page
Stable Determination of the Electromagnetic Coefficients by Boundary Measurements
The goal of this paper is to prove a stable determination of the coefficients
for the time-harmonic Maxwell equations, in a Lipschitz domain, by boundary
measurements
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