53 research outputs found
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 - or -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 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
- …