17,968 research outputs found
Eigenfunction localization for the 2D periodic Schr\"odinger operator
We prove that for any {\it fixed} trigonometric polynomial potential
satisfying a genericity condition, the spectrum of the two dimension periodic
Schr\"odinger operator has finite multiplicity and the Fourier series of the
eigenfunctions are uniformly exponentially localized about a finite number of
frequencies. As a corollary, the norms of the eigenfunctions are bounded
for all , which answers a question of Toth and Zelditch \cite{TZ}.Comment: 26 p
Spectral Methods in PDE
This is to review some recent progress in PDE. The emphasis is on (energy)
supercritical nonlinear Schr\"odinger equations. The methods are applicable to
other nonlinear equations.Comment: This is an invited contribution to Milan J. Math., after a talk at
the Seminario Matematico e Fisico di Milano. It also contains a new result on
critical Sobolev exponent for the cubic nonlinear Schr\"odinger equation on
general surfaces. (13pp
Integrals of products of Hermite functions
We compute the integrals of products of Hermite functions using the
generating functions. The precise asymptotics of 4 Hermite functions are
presented below. This estimate is relevant for the corresponding cubic
nonlinear equation.Comment: 8 p
Quasi-periodic solutions of the Schr\"odinger equation with arbitrary algebraic nonlinearities
We present a geometric formulation of existence of time quasi-periodic
solutions. As an application, we prove the existence of quasi-periodic
solutions of frequencies, , in arbitrary dimension and for
arbitrary non integrable algebraic nonlinearity . This reflects the
conservation of momenta, energy and norm. In 1d, we prove the
existence of quasi-periodic solutions with arbitrary and for arbitrary ,
solving a problem that started Hamiltonian PDE.Comment: 19 pp, slightly more details on proofs of Theorems 2 and
Quasi-periodic solutions for nonlinear wave equations (announcement)
We construct time quasi-periodic solutions to nonlinear wave equations on the
torus in arbitrary dimensions. All previously known results (in the case of
zero or a multiplicative potential) seem to be limited to the circle. This
generalizes the method developed in the limit-elliptic setting of NLS to the
hyperbolic setting. The additional ingredient is a Diophantine property of
algebraic numbers.Comment: Completely different from the initial, 2011 versio
Anderson Localization for Time Quasi Periodic Random Sch\"odinger and Wave Operators
We prove that at large disorder, with large probability and for a set of
Diophantine frequencies of large measure, Anderson localization in
is {\it stable} under localized time-quasi-periodic perturbations by proving
that the associated quasi-energy operator has pure point spectrum. The main
tools are the Fr\"ohlich-Spencer mechanism for the random component and the
Bourgain-Goldstein-Schlag mechanism for the quasi-periodic component. The
formulation of this problem is motivated by questions of Anderson localization
for non-linear Schr\"odinger equations.Comment: 37 pg
Anderson Localization for Time Periodic Random Schr\"odinger operators
We prove that at large disorder, Anderson localization in is stable
under localized time-periodic perturbations by proving that the associated
quasi-energy operator has pure point spectrum. The formulation of this problem
is motivated by questions of Anderson localization for non-linear Schr\"odinger
equations.Comment: 15 pgs, to appear in Commun PD
Strong rainbow connection numbers of toroidal meshes
In 2011, Li et al. \cite{LLL} obtained an upper bound of the strong rainbow
connection number of an -dimensional undirected toroidal mesh. In this
paper, this bound is improved. As a result, we give a negative answer to their
problem.Comment: 9 pages, 3 figure
Pseudo-potential of a power-law decaying interaction in two-dimensional systems
We analytically derive the general pseudo-potential operator of an arbitrary
isotropic interaction for particles confined in two-dimensional (2D) systems,
using the frame work developed by Huang and Yang for 3D scattering. We also
analytically derive the low energy dependence of the scattering phase-shift for
an arbitrary interaction with a power-law decaying tail, (for ). We apply our results to the
2D dipolar gases () as an example, calculating the momentum and
dipole moment dependence of the pseudo-potential for both - and p-wave
scattering channels if the two scattering particles are in the same 2D layer.
Results for the s-wave scattering between particles in two different (parallel)
layers are also investigated. Our results can be directly applied to the
systems of dipolar atoms and/or polar molecules in a general 2D geometry.Comment: 4 pages and 2 figures. Some correction is made for the
pseudo-potential of higher angular momentum. Mre references are adde
Fermion Masses and Flavor Mixing in A Supersymmetric SO(10) Model
we study fermion masses and flavor mixing in a supersymmetric SO(10) model,
where , and Higgs multiplets
have Yukawa couplings with matter multiplets and give masses to quarks and
leptons through the breaking chain of a Pati-Salam group. This brings about
that, at the GUT energy scale, the lepton mass matrices are related to the
quark ones via several breaking parameters, and the small neutrino masses arise
from a Type II see-saw mechanism. When evolving renormalization group equations
for the fermion mass matrices from the GUT scale to the electroweak scale, in a
specific parameter scenario, we show that the model can elegantly accommodate
all observed values of masses and mixing for the quarks and leptons,
especially, it's predictions for the bi-large mixing in the leptonic sector are
very well in agreement with the current neutrino experimental data.Comment: LaTeX2e, 14 pages, 2 figure
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