1,281 research outputs found
Graph-like asymptotics for the Dirichlet Laplacian in connected tubular domains
We consider the Dirichlet Laplacian in a waveguide of uniform width and
infinite length which is ideally divided into three parts: a "vertex region",
compactly supported and with non zero curvature, and two "edge regions" which
are semi-infinite straight strips. We make the waveguide collapse onto a graph
by squeezing the edge regions to half-lines and the vertex region to a point.
In a setting in which the ratio between the width of the waveguide and the
longitudinal extension of the vertex region goes to zero, we prove the
convergence of the operator to a selfadjoint realization of the Laplacian on a
two edged graph. In the limit operator, the boundary conditions in the vertex
depend on the spectral properties of an effective one dimensional Hamiltonian
associated to the vertex region.Comment: Major revision. Reviewed introduction. Changes in Th. 1, Th. 2, and
Th. 3. Updated references. 23 page
Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide
In distinction to the Neumann case the squeezing limit of a Dirichlet network
leads in the threshold region generically to a quantum graph with disconnected
edges, exceptions may come from threshold resonances. Our main point in this
paper is to show that modifying locally the geometry we can achieve in the
limit a nontrivial coupling between the edges including, in particular, the
class of -type boundary conditions. We work out an illustration of this
claim in the simplest case when a bent waveguide is squeezed.Comment: LaTeX, 16 page
Relative partition function of Coulomb plus delta interaction
The relative partition function and the relative zeta function of the
perturbation of the Laplace operator by a Coulomb potential plus a point
interaction centered in the origin is discussed. Applications to the study of
the Casimir effect are indicated.Comment: Minor misprints corrected. 24 page
Quasi-1D Bose-Einstein condensates in the dimensional crossover regime
We study theoretically the dimensional crossover from a three-dimensional
elongated condensate to a one-dimensional condensate as the transverse degrees
of freedom get frozen by tight confinement, in the limit of small density
fluctuations, i.e. for a strongly degenerate gas. We compute analytically the
radially integrated density profile at low temperatures using a local density
approximation, and study the behavior of phase fluctuations with the transverse
confinement. Previous studies of phase fluctuations in trapped gases have
either focused on the 3D elongated regimes or on the 1D regime. The present
approach recovers these previous results and is able to interpolate between
them. We show in particular that in this strongly degenerate limit the shape of
the spatial correlation function is insensitive to the transverse regime of
confinement, pointing out to an almost universal behavior of phase fluctuations
in elongated traps
Time dependent delta-prime interactions in dimension one
We solve the Cauchy problem for the Schr\"odinger equation corresponding to
the family of Hamiltonians in which
describes a -interaction with time-dependent strength .
We prove that the strong solution of such a Cauchy problem exits whenever the
map belongs to the fractional Sobolev space
, thus weakening the hypotheses which would be required by
the known general abstract results. The solution is expressed in terms of the
free evolution and the solution of a Volterra integral equation.Comment: minor changes, 10 page
Effective equation for a system of mechanical oscillators in an acoustic field
We consider a one dimensional evolution problem modeling the dynamics of an
acoustic field coupled with a set of mechanical oscillators. We analyze
solutions of the system of ordinary and partial differential equations with
time-dependent boundary conditions describing the evolution in the limit of a
continuous distribution of oscillators.Comment: Improved Theorem 2. Updated introduction and references. Added 1
figure. 11 page
Towards a Distributed Quantum Computing Ecosystem
The Quantum Internet, by enabling quantum communications among remote quantum
nodes, is a network capable of supporting functionalities with no direct
counterpart in the classical world. Indeed, with the network and communications
functionalities provided by the Quantum Internet, remote quantum devices can
communicate and cooperate for solving challenging computational tasks by
adopting a distributed computing approach. The aim of this paper is to provide
the reader with an overview about the main challenges and open problems arising
with the design of a Distributed Quantum Computing ecosystem. For this, we
provide a survey, following a bottom-up approach, from a communications
engineering perspective. We start by introducing the Quantum Internet as the
fundamental underlying infrastructure of the Distributed Quantum Computing
ecosystem. Then we go further, by elaborating on a high-level system
abstraction of the Distributed Quantum Computing ecosystem. Such an abstraction
is described through a set of logical layers. Thereby, we clarify dependencies
among the aforementioned layers and, at the same time, a road-map emerges
On the structure of critical energy levels for the cubic focusing NLS on star graphs
We provide information on a non trivial structure of phase space of the cubic
NLS on a three-edge star graph. We prove that, contrarily to the case of the
standard NLS on the line, the energy associated to the cubic focusing
Schr\"odinger equation on the three-edge star graph with a free (Kirchhoff)
vertex does not attain a minimum value on any sphere of constant -norm. We
moreover show that the only stationary state with prescribed L^2-norm is indeed
a saddle point
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