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 $L^p$ norms of the eigenfunctions are bounded for all $p>0$, 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 $b$ frequencies, $b\leq d+2$, in arbitrary dimension $d$ and for arbitrary non integrable algebraic nonlinearity $p$. This reflects the conservation of $d$ momenta, energy and $L^2$ norm. In 1d, we prove the existence of quasi-periodic solutions with arbitrary $b$ and for arbitrary $p$, 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 $\Bbb Z^d$ 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 $\Z^d$ 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 $r$-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, $V_{\rm 2D}(\rho)\propto \rho^{-\alpha}$ (for $\alpha>2$). We apply our results to the 2D dipolar gases ($\alpha=3$) as an example, calculating the momentum and dipole moment dependence of the pseudo-potential for both $s$- 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 $\mathbf{10}$, $\mathbf{120}$ and $\mathbf{\bar{126}}$ 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