Defect Modes and Homogenization of Periodic Schr\"odinger Operators

We consider the discrete eigenvalues of the operator H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x), where V(\x) is periodic and Q(\y) is localized on $\R^d,\ \ d\ge1$. For \eps>0 and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schr\"odinger operator, H_0 = -\Delta_\x+V(\x), into spectral gaps. The nature of the bifurcation depends on the homogenized Schr\"odinger operator L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y). Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of the bifurcation.Comment: 26 pages, 3 figures, to appear SIAM J. Math. Ana

Aerodynamic heating in the vicinity of corners at hypersonic speeds

Aerodynamic heating in vicinity of corners at hypersonic speed

Symplectic Microgeometry II: Generating functions

We adapt the notion of generating functions for lagrangian submanifolds to symplectic microgeometry. We show that a symplectic micromorphism always admits a global generating function. As an application, we describe hamiltonian flows as special symplectic micromorphisms whose local generating functions are the solutions of Hamilton-Jacobi equations. We obtain a purely categorical formulation of the temporal evolution in classical mechanics.Comment: 27 pages, 1 figur

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model

General boundary conditions ("branes") for the Poisson sigma model are studied. They turn out to be labeled by coisotropic submanifolds of the given Poisson manifold. The role played by these boundary conditions both at the classical and at the perturbative quantum level is discussed. It turns out to be related at the classical level to the category of Poisson manifolds with dual pairs as morphisms and at the perturbative quantum level to the category of associative algebras (deforming algebras of functions on Poisson manifolds) with bimodules as morphisms. Possibly singular Poisson manifolds arising from reduction enter naturally into the picture and, in particular, the construction yields (under certain assumptions) their deformation quantization.Comment: 21 pages, 2 figures; minor corrections, references updated; final versio

Sub-Natural-Linewidth Quantum Interference Features Observed in Photoassociation of a Thermal Gas

By driving photoassociation transitions we form electronically excited molecules (Na$_2^*$) from ultra-cold (50-300 $\mu$K) Na atoms. Using a second laser to drive transitions from the excited state to a level in the molecular ground state, we are able to split the photoassociation line and observe features with a width smaller than the natural linewidth of the excited molecular state. The quantum interference which gives rise to this effect is analogous to that which leads to electromagnetically induced transparency in three level atomic $\Lambda$ systems, but here one of the ground states is a pair of free atoms while the other is a bound molecule. The linewidth is limited primarily by the finite temperature of the atoms.Comment: 4 pages, 5 figure