408 research outputs found
Perturbations of eigenvalues embedded at threshold: one, two and three dimensional solvable models
We examine perturbations of eigenvalues and resonances for a class of
multi-channel quantum mechanical model-Hamiltonians describing a particle
interacting with a localized spin in dimension . We consider
unperturbed Hamiltonians showing eigenvalues and resonances at the threshold of
the continuous spectrum and we analyze the effect of various type of
perturbations on the spectral singularities. We provide algorithms to obtain
convergent series expansions for the coordinates of the singularities.Comment: 20 page
Constrained energy minimization and orbital stability for the NLS equation on a star graph
We consider a nonlinear Schr\"odinger equation with focusing nonlinearity of
power type on a star graph , written as , where is the selfadjoint operator
which defines the linear dynamics on the graph with an attractive
interaction, with strength , at the vertex. The mass and energy
functionals are conserved by the flow. We show that for the energy at
fixed mass is bounded from below and that for every mass below a critical
mass it attains its minimum value at a certain \hat \Psi_m \in H^1(\GG)
, while for there is no minimum. Moreover, the set of minimizers has
the structure {\mathcal M}={e^{i\theta}\hat \Psi_m, \theta\in \erre}.
Correspondingly, for every there exists a unique
such that the standing wave is orbitally
stable. To prove the above results we adapt the concentration-compactness
method to the case of a star graph. This is non trivial due to the lack of
translational symmetry of the set supporting the dynamics, i.e. the graph. This
affects in an essential way the proof and the statement of
concentration-compactness lemma and its application to minimization of
constrained energy. The existence of a mass threshold comes from the
instability of the system in the free (or Kirchhoff's) case, that in our
setting corresponds to \al=0.Comment: 26 pages, 1 figur
Variational properties and orbital stability of standing waves for NLS equation on a star graph
We study standing waves for a nonlinear Schr\"odinger equation on a star
graph {} i.e. half-lines joined at a vertex. At the vertex an
interaction occurs described by a boundary condition of delta type with
strength . The nonlinearity is of focusing power type. The
dynamics is given by an equation of the form , where is the Hamiltonian operator which
generates the linear Schr\"odinger dynamics. We show the existence of several
families of standing waves for every sign of the coupling at the vertex for
every . Furthermore, we determine the ground
states, as minimizers of the action on the Nehari manifold, and order the
various families. Finally, we show that the ground states are orbitally stable
for every allowed if the nonlinearity is subcritical or critical, and
for otherwise.Comment: 36 pages, 2 figures, final version appeared in JD
The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit
In the present paper we study the following scaled nonlinear Schr\"odinger
equation (NLS) in one space dimension: This equation represents a nonlinear Schr\"odinger
equation with a spatially concentrated nonlinearity. We show that in the limit
, the weak (integral) dynamics converges in to
the weak dynamics of the NLS with point-concentrated nonlinearity: where is the
laplacian with the nonlinear boundary condition at the origin
and
. The convergence occurs for every if and for every otherwise. The same
result holds true for a nonlinearity with an arbitrary number of
concentration pointsComment: 10 page
Quantum test of the equivalence principle for atoms in superpositions of internal energy eigenstates
The Einstein Equivalence Principle (EEP) has a central role in the
understanding of gravity and space-time. In its weak form, or Weak Equivalence
Principle (WEP), it directly implies equivalence between inertial and
gravitational mass. Verifying this principle in a regime where the relevant
properties of the test body must be described by quantum theory has profound
implications. Here we report on a novel WEP test for atoms. A Bragg atom
interferometer in a gravity gradiometer configuration compares the free fall of
rubidium atoms prepared in two hyperfine states and in their coherent
superposition. The use of the superposition state allows testing genuine
quantum aspects of EEP with no classical analogue, which have remained
completely unexplored so far. In addition, we measure the Eotvos ratio of atoms
in two hyperfine levels with relative uncertainty in the low ,
improving previous results by almost two orders of magnitude.Comment: Accepted for publication in Nature Communicatio
Photodynamic therapy: a treatment tool for hidradenitis suppurativa
Hidradenitis suppurativa (HS) is a chronic, recurrent, debilitating inflammatory disease of unknown aetiology. The current standard of care for HS includes antibiotics (oral/topical), retinoids (oral/topical) and intralesional steroids. However, the recurrence of the disease and the unsuccessful of the different treatments stimulate findings of different therapy using light source. Among these, photodynamic therapy with 5-aminolevulinic acid (ALA-PDT) has been proposed by some authors recently. In this study we report the experience of 15 patients affected by HS resistant to conventional therapy treated with ALA-PDT
Point interactions in acoustics: one dimensional models
A one dimensional system made up of a compressible fluid and several
mechanical oscillators, coupled to the acoustic field in the fluid, is analyzed
for different settings of the oscillators array. The dynamical models are
formulated in terms of singular perturbations of the decoupled dynamics of the
acoustic field and the mechanical oscillators. Detailed spectral properties of
the generators of the dynamics are given for each model we consider. In the
case of a periodic array of mechanical oscillators it is shown that the energy
spectrum presents a band structure.Comment: revised version, 30 pages, 2 figure
On the spectrum of a bent chain graph
We study Schr\"odinger operators on an infinite quantum graph of a chain form
which consists of identical rings connected at the touching points by
-couplings with a parameter . If the graph is "straight",
i.e. periodic with respect to ring shifts, its Hamiltonian has a band spectrum
with all the gaps open whenever . We consider a "bending"
deformation of the chain consisting of changing one position at a single ring
and show that it gives rise to eigenvalues in the open spectral gaps. We
analyze dependence of these eigenvalues on the coupling and the
"bending angle" as well as resonances of the system coming from the bending. We
also discuss the behaviour of the eigenvalues and resonances at the edges of
the spectral bands.Comment: LaTeX, 23 pages with 7 figures; minor changes, references added; to
appear in J. Phys. A: Math. Theo
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