Wigner Functions versus WKB-Methods in Multivalued Geometrical Optics

We consider the Cauchy-problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of high-frequency asymptotics of such models is reviewed,in particular we highlight the difficulties in crossing caustics when using (time-dependent) WKB-methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono-kinetic solutions) in the prebreaking regime. Further we show that the Wigner measure approach can be used to analyze high-frequency limits in the post-breaking regime, in comparison with the traditional Fourier integral operator method. Finally we present some illustrating examples.Comment: 38 page

Optimal bilinear control of Gross-Pitaevskii equations

A mathematical framework for optimal bilinear control of nonlinear Schr\"odinger equations of Gross-Pitaevskii type arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used $L^2$- or $H^1$-norms for the cost of the control action. Well-posedness of the problem and existence of an optimal control is proven. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.Comment: 30 pages, 14 figure

On the dynamics of Bohmian measures

International audienceWe revisit the concept of Bohmian measures recently introduced by the authors in \cite{MPS}. We rigorously prove that for sufficiently smooth wave functions the corresponding Bohmian measure furnishes a distributional solution of a nonlinear Vlasov-type equation. Moreover, we study the associated defect measures appearing in the classical limit. In one space dimension, this yields a new connection between mono-kinetic Wigner and Bohmian measures. In addition, we shall study the dynamics of Bohmian measures associated to so-called semi-classical wave packets. For these type of wave functions, we prove local in-measure convergence of a rescaled sequence of Bohmian trajectories towards the classical Hamiltonian flow on phase space. Finally, we construct an example of wave functions whose limiting Bohmian measure is not mono-kinetic but nevertheless equals the associated Wigner measure

A time-splitting spectral scheme for the Maxwell-Dirac system

We present a time-splitting spectral scheme for the Maxwell-Dirac system and similar time-splitting methods for the corresponding asymptotic problems in the semi-classical and the non-relativistic regimes. The scheme for the Maxwell-Dirac system conserves the Lorentz gauge condition, is unconditionally stable and highly efficient as our numerical examples show. In particular we focus in our examples on the creation of positronic modes in the semi-classical regime and on the electron-positron interaction in the non-relativistic regime. Furthermore, in the non-relativistic regime, our numerical method exhibits uniform convergence in the small parameter \dt, which is the ratio of the characteristic speed and the speed of light.Comment: 29 pages, 119 figure

On a dissipative Gross-Pitaevskii-type model for exciton-polariton condensates

We study a generalized dissipative Gross-Pitaevskii-type model arising in the description of exciton-polariton condensates. We derive global in-time existence results and various a-priori estimates for this model posed on the one-dimensional torus. Moreover, we analyze in detail the long-time behavior of spatially homogenous solutions and their respective steady states and present numerical simulations in the case of more general initial data. We also study the convergence to the corresponding adiabatic regime, which results in a single damped-driven Gross-Pitaveskii equation.Comment: 25 pages, 11 figure

Classical limit for semi-relativistic Hartree systems

We consider the three-dimensional semi-relativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove, using Wigner transformation techniques, that its classical limit yields the well known relativistic Vlasov-Poisson system. The result holds for the case of attractive and repulsive mean-field interaction, with an additional size constraint in the attractive case.Comment: 10 page

(Semi)classical limit of the Hartree equation with harmonic potential

Nonlinear Schrodinger Equations (NLS) of the Hartree type occur in the modeling of quantum semiconductor devices. Their "semiclassical" limit of vanishing (scaled) Planck constant is both a mathematical challenge and practically relevant when coupling quantum models to classical models. With the aim of describing the semi-classical limit of the 3D Schrodinger--Poisson system with an additional harmonic potential, we study some semi-classical limits of the Hartree equation with harmonic potential in space dimension n>1. The harmonic potential is confining, and causes focusing periodically in time. We prove asymptotics in several cases, showing different possible nonlinear phenomena according to the interplay of the size of the initial data and the power of the Hartree potential. In the case of the 3D Schrodinger-Poisson system with harmonic potential, we can only give a formal computation since the need of modified scattering operators for this long range scattering case goes beyond current theory. We also deal with the case of an additional "local" nonlinearity given by a power of the local density - a model that is relevant when incorporating the Pauli principle in the simplest model given by the "Schrodinger-Poisson-X$\alpha$ equation". Further we discuss the connection of our WKB based analysis to the Wigner function approach to semiclassical limits.Comment: 26 page

On the Gross-Pitaevskii equation for trapped dipolar quantum gases

We study the time-dependent Gross-Pitaevskii equation describing Bose-Einstein condensation of trapped dipolar quantum gases. Existence and uniqueness as well as the possible blow-up of solutions are studied. Moreover, we discuss the problem of dimension-reduction for this nonlinear and nonlocal Schrodinger equation.Comment: 21 page

Haploidentical hematopoietic stem cell transplantation as individual treatment option in pediatric patients with very high-risk sarcomas

Background Prognosis of children with primary disseminated or metastatic relapsed sarcomas remains dismal despite intensification of conventional therapies including high-dose chemotherapy. Since haploidentical hematopoietic stem cell transplantation (haplo-HSCT) is effective in the treatment of hematological malignancies by mediating a graft versus leukemia effect, we evaluated this approach in pediatric sarcomas as well. Methods Patients with bone Ewing sarcoma or soft tissue sarcoma who received haplo-HSCT as part of clinical trials using CD3+ or TCRα/β+ and CD19+ depletion respectively were evaluated regarding feasibility of treatment and survival. Results We identified 15 patients with primary disseminated disease and 14 with metastatic relapse who were transplanted from a haploidentical donor to improve prognosis. Three-year event-free survival (EFS) was 18,1% and predominantly determined by disease relapse. Survival depended on response to pre-transplant therapy (3y-EFS of patients in complete or very good partial response: 36,4%). However, no patient with metastatic relapse could be rescued. Conclusion Haplo-HSCT for consolidation after conventional therapy seems to be of interest for some, but not for the majority of patients with high-risk pediatric sarcomas. Evaluation of its future use as basis for subsequent humoral or cellular immunotherapies is necessary