430 research outputs found
The Flux-Across-Surfaces Theorem for a Point Interaction Hamiltonian
The flux-across-surfaces theorem establishes a fundamental relation in
quantum scattering theory between the asymptotic outgoing state and a quantity
which is directly measured in experiments. We prove it for a hamiltonian with a
point interaction, using the explicit expression for the propagator. The proof
requires only assuptions on the initial state and it covers also the case of
zero-energy resonance. We also outline a different approach based on
generalized eigenfunctions, in view of a possible extension of the result.Comment: AMS-Latex file, 11 page
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
Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on S2 . Precisely, local well-posedness is proved for any C 2 power-nonlinearity, while global well-posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point-concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas
A Quantum Model of Feshbach Resonances
We consider a quantum model of two-channel scattering to describe the
mechanism of a Feshbach resonance. We perform a rigorous analysis in order to
count and localize the energy resonances in the perturbative regime, i.e., for
small inter-channel coupling, and in the non-perturbative one. We provide an
expansion of the effective scattering length near the resonances, via a
detailed study of an effective Lippmann-Schwinger equation with
energy-dependent potential.Comment: 29 pages, pdfLaTe
Cholesterol derivatives make large part of the lipids from epidermal molts of the desert-adapted Gila monster lizard (Heloderma suspectum)
In order to understand the cutaneous water loss in the desert-adapted and venomous lizard Heloderma suspectum, the microscopic structure and lipid composition of epidermal molts have been examined using microscopic, spectroscopic and chemical analysis techniques. The molt is formed by a variably thick, superficial beta-layer, an extensive mesos-region and few alpha-cells in its lowermost layers. The beta-layer contains most corneous beta proteins while the mesos-region is much richer in lipids. The proteins in the mesos-region are more unstructured than those located in the beta-layer. Most interestingly, among other lipids, high contents of cholesteryl-β-glucoside and cholesteryl sulfate were detected, molecules absent or present in traces in other species of squamates. These cholesterol derivatives may be involved in the stabilization and compaction of the mesos-region, but present a limited permeability to water movements. The modest resistance to cutaneous water-loss of this species is compensated by adopting other physiological strategies to limit thermal damage and water transpiration as previous eco-physiological studies have indicated. The increase of steroid derivatives may also be implicated in the heat shock response, influencing the relative behavior in this desert-adapted lizard
Wave equation with concentrated nonlinearities
In this paper we address the problem of wave dynamics in presence of
concentrated nonlinearities. Given a vector field on an open subset of
\CO^n and a discrete set Y\subset\RE^3 with elements, we define a
nonlinear operator on L^2(\RE^3) which coincides with the free
Laplacian when restricted to regular functions vanishing at , and which
reduces to the usual Laplacian with point interactions placed at when
is linear and is represented by an Hermitean matrix. We then consider the
nonlinear wave equation and study the
corresponding Cauchy problem, giving an existence and uniqueness result in the
case is Lipschitz. The solution of such a problem is explicitly expressed
in terms of the solutions of two Cauchy problem: one relative to a free wave
equation and the other relative to an inhomogeneous ordinary differential
equation with delay and principal part . Main properties of
the solution are given and, when is a singleton, the mechanism and details
of blow-up are studied.Comment: Revised version. To appear in Journal of Physics A: Mathematical and
General, special issue on Singular Interactions in Quantum Mechanics:
Solvable Model
Boundary Conditions for Singular Perturbations of Self-Adjoint Operators
Let A:D(A)\subseteq\H\to\H be an injective self-adjoint operator and let
\tau:D(A)\to\X, X a Banach space, be a surjective linear map such that
\|\tau\phi\|_\X\le c \|A\phi\|_\H. Supposing that \text{\rm Range}
(\tau')\cap\H' =\{0\}, we define a family of self-adjoint
operators which are extensions of the symmetric operator .
Any in the operator domain is characterized by a sort
of boundary conditions on its univocally defined regular component \phireg,
which belongs to the completion of D(A) w.r.t. the norm \|A\phi\|_\H. These
boundary conditions are written in terms of the map , playing the role of
a trace (restriction) operator, as \tau\phireg=\Theta Q_\phi, the extension
parameter being a self-adjoint operator from X' to X. The self-adjoint
extension is then simply defined by A^\tau_\Theta\phi:=A \phireg. The case in
which is a convolution operator on LD, T a distribution with
compact support, is studied in detail.Comment: Revised version. To appear in Operator Theory: Advances and
Applications, vol. 13
Decay of a Bound State under a Time-Periodic Perturbation: a Toy Case
We study the time evolution of a three dimensional quantum particle,
initially in a bound state, under the action of a time-periodic zero range
interaction with ``strength'' (\alpha(t)). Under very weak generic conditions
on the Fourier coefficients of (\alpha(t)), we prove complete ionization as (t
\to \infty). We prove also that, under the same conditions, all the states of
the system are scattering states.Comment: LaTeX2e, 15 page
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