Semiclassical evolution with low regularity

Semiclassical Second Microlocal Propagation of Regularity and Integrable Systems

We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold X with respect to a Lagrangian submanifold of T X. The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the con...

Low-regularity Schrödinger Maps

We prove that the Schrödinger map initial-value problem { ∂ts = s×∆xs on R × [−1, 1]; s(0) = s0 is locally well-posed for small data s0 ∈ H σ0 Q (R ; S), σ0 > (d+ 1)/2, Q ∈ S.

Maximal Regularity for Evolution Equations

Semiclassical Evolution of Dissipative Markovian Systems

A semiclassical approximation for an evolving density operator, driven by a “closed” hamiltonian operator and “open” markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the hamiltonian operator is a quadratic function and the Lindblad operators ...

On the semiclassical evolution of quantum operators

The Heisenberg evolution of a given unitary operator corresponds classically to a fixed canonical transformation that is viewed through a moving coordinate system. The operators that form the bases of the Weyl representation and its Fourier transform, the chord representation, are, respectively, unitary reflection and translation operators. Thus, the general semiclassical study of unitary opera...