We study the homogenization of a Schrödinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous ”Fermi golden rule” for th...

A nonlinear generalisation of Schrodinger’s equation had previously been obtained using information-theoretic arguments. The nonlinearities in that equation were of a nonpolynomial form, equivalent to the occurence of higher-derivative nonlinear terms at all orders. Here we construct some exact solutions to that equation in 1 + 1 dimensions. On the half-line, the solutions resemble exponentiall...

The belief propagation algorithm has been recognized in the information theory community as a soft-decision iterative decoding algorithm. It is the most powerful algorithm found so far for attacking hard optimization problems in channel decoding. Quantum mechanics is the foundation of modern physics with the time-independent Schrödinger equation being one of the most important equations. This p...

We prove sharp Strichartz estimates for the semi-classical Schrödinger equation on a compact Riemannian manifold with smooth, strictly geodesically concave boundary. We deduce classical Strichartz estimates for the Schrödinger equation outside a strictly convex obstacle, local existence for the H1-critical (quintic) Schrödinger equation and scattering for the sub-critical Schrödinger equation i...

The disordered Bose Hubbard model is studied numerically within the Bogoliubov approximation. First, the spatially varying condensate wavefunction in the presence of disorder is found by solving a nonlinear Schrodinger equation. Using the Bogoliubov approximation to find the excitations above this condensate, we calculate the condensate fraction, superfluid density, and density of states for a ...

We consider neutrino oscillations as non stationary phenomenon based on Schrodinger evolution equation and mixed neutrino states with definite flavor. We demonstrate that for such states invariance under translations in time does not take place. We show that time-energy uncertainty relation plays a crucial role in neutrino oscillations. We compare neutrino oscillations with K 0 ⇆ ¯ K 0 , B 0 d ...

We study the reducibility of a linear Schrodinger equation subject to small unbounded almost periodic perturbation which is analytic in time and space. Under appropriate assumptions on smalln...