Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSemi-classical measures on Quantum Graphs and the Gauss map of the determinant manifold
101
0
0.0
(
0
)
Added by
Yves Colin de Verdiere
Publication date
2013
fields
Physics
and research's language is
English
Authors
Yves Colin De Verdi`ere
Ask ChatGPT about the research
No Arabic abstract
In this paper, I describe the weak limits of the measures associated to the eigenfunctions of the Laplacian on a Quantum graph for a generic metric in terms of the Gauss map of the determinant manifold. I describe also all the limits with minimal support (the scars).
rate research
Read More
We introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph.
begin{abstract} We show that if the initial profile $qleft( xright) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $int^{infty }x^{5/2}leftvert qleft( xright) rightvert dx<infty,$ (no decay at $-infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. end{abstract}
We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincare equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm, and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with $2$-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.
The fully compressible semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alternative proof of the earlier result on the existence of weak solutions of this system expressed in the so-called geostrophic, or dual, coordinates. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measures.
We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian $mathsf H$ is given, as sum of quadratic forms, by $mathsf H= -frac{hbar^{2}}{2m},frac{d^{2},}{dx^{2}},dot{+},alphadelta_{0}$, with $alphainmathbb R$ and $delta_{0}$ the Dirac delta-distribution at $x=0$. We show that the quantum evolution can be approximated, uniformly for any time away from the collision time and with an error of order $hbar^{3/2-lambda}$, $0!<!lambda!<!3/2$, by the quasi-classical evolution generated by a self-adjoint extension of the restriction to $mathcal C^{infty}_{c}({mathscr M}_{0})$, ${mathscr M}_{0}:={(q,p)!in!mathbb R^{2},|,q! ot=!0}$, of ($-i$ times) the generator of the free classical dynamics; such a self-adjoint extension does not correspond to the classical dynamics describing the complete reflection due to the infinite barrier. Similar approximation results are also provided for the wave and scattering operators.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
