We consider short-range perturbations of elliptic operators on $R^d$ with constant coefficients, and study the asymptotic properties of the scattering matrix as the energy tends to infinity. We give the leading terms of the symbol of the scattering matrix. The proof employs semiclassical analysis combined with a generalization of the Isozaki-Kitada theory on time-independent modifiers. We also consider scattering matrices for 2 and 3 dimensional Dirac operators. (joint work with Alexander Pushnitski (King's College London)[-]

We present the Quantum Vlasov or Wigner equation as a "phase space" presentation of quantum mechanics that is close to the classical Vlasov equation, but where the "distribution function" $w(x,v,t)$ will in general have also negative values.
We discuss the relation to the classical Vlasov equation in the semi-classical asymptotics of small Planck's constant, for the linear case [2] and for the nonlinear case where we couple the quantum Vlasov equation to the Poisson equation [4, 3, 5] and [1].
Recently, in some sort of "inverse semiclassical limit" the numerical concept of solving Schrödinger-Poisson as an approximation of Vlasov-Poisson attracted attention in cosmology, which opens a link to the "smoothed Schrödinger/Wigner numerics" of Athanassoulis et al. (e.g. [6]).[-]

One-particle density matrix is the key object in the quantum-mechanical approximation schemes. In this talk I will give a short survey of recent regularity results with emphasis on sharp bounds for the eigenfunctions, and show how these bounds lead to the asymptotic formula for the eigenvalues of the one-particle density matrix. The argument is based on the results of M. Birman and M. Solomyak on spectral asymptotics for pseudo-differential operators with matrix-valued symbols.[-]

We consider the operator $\mathcal{A}_h = -h^2 \Delta + iV$ in the semi-classical limit $h \to 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of $\mathcal{A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.[-]

We discuss the 2D Schrödinger equation for periodic potentials with the symmetry of a hexagonal tiling of the plane. We first review joint work with CL Fefferman on the existence of Dirac points, conical singularities in the band structure, and the resulting effective 2D Dirac dynamics of wave-packets. We then focus on periodic potentials which are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure, corresponding to the single electron model of graphene and its artificial analogues. We prove that for sufficiently deep potentials (strong binding) the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, are uniformly close to those of the celebrated two-band tight-binding model, introduced by PR Wallace (1947) in his pioneering study of graphite. We then discuss corollaries, in the strong binding regime, on (a) spectral gaps for honeycomb potentials with PT symmetry-breaking perturbations, and (b) topologically protected edge states for honeycomb structures with "rational edges. This is joint work with CL Fefferman and JP Lee-Thorp. Extensions to Maxwell equations (with Y Zhu and JP Lee-Thorp) will also be discussed.[-]

We study the survival probability associated with a semiclassical matrix Schrödinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual time-dependent exponential contribution, up to a reminder term that is small in the semiclassical parameter and for which we find the main contribution. The result applies in any dimension, and in presence of a number of resonances that may tend to infinity as the semiclassical parameter tends to 0.
This is a joint work with Ph. Briet.[-]