Green's functions for the schrödinger operator with periodic potential

Modulational Instability for Nonlinear Schrödinger Equations with a Periodic Potential

We study the stability properties of periodic solutions to the Nonlinear Schrödinger (NLS) equation with a periodic potential. We exploit the symmetries of the problem, in particular the Hamiltonian structure and the U(1) symmetry. We develop a simple sufficient criterion that guarantees the existence of a modulational instability spectrum along the imaginary axis. In the case of small amplitud...

Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential.

In the present paper we use the Wannier function basis to construct lattice approximations of the nonlinear Schrödinger equation with a periodic potential. We show that the nonlinear Schrödinger equation with a periodic potential is equivalent to a vector lattice with long-range interactions. For the case-example of the cosine potential we study the validity of the so-called tight-binding appro...

Semiclassical reduction for magnetic Schrödinger operator with periodic zero-range potentials and applications

The two-dimensional Schrödinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields and a weak coupling we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like operators. This shows the existence of parts of Cantor structure in the spectrum for special values of the magnetic flux.

Ground State for the Schrödinger Operator with the Weighted Hardy Potential

Greens Functions for the Wave Equation

I gather together known results on fundamental solutions to the wave equation in free space, and Greens functions in tori, boxes, and other domains. From this the corresponding fundamental solutions for the Helmholtz equation are derived, and, for the 2D case the semiclassical approximation interpreted back in the time-domain. Utility: scarring via time-dependent propagation in cavities; Math 4...