15,742 research outputs found
Approximation of Feynman path integrals with non-smooth potentials
We study the convergence in of the time slicing approximation of
Feynman path integrals under low regularity assumptions on the potential.
Inspired by the custom in Physics and Chemistry, the approximate propagators
considered here arise from a series expansion of the action. The results are
ultimately based on function spaces, tools and strategies which are typical of
Harmonic and Time-frequency analysis.Comment: 18 page
Linear-Quadratic -person and Mean-Field Games with Ergodic Cost
We consider stochastic differential games with players, linear-Gaussian
dynamics in arbitrary state-space dimension, and long-time-average cost with
quadratic running cost. Admissible controls are feedbacks for which the system
is ergodic. We first study the existence of affine Nash equilibria by means of
an associated system of Hamilton-Jacobi-Bellman and
Kolmogorov-Fokker-Planck partial differential equations. We give necessary and
sufficient conditions for the existence and uniqueness of quadratic-Gaussian
solutions in terms of the solvability of suitable algebraic Riccati and
Sylvester equations. Under a symmetry condition on the running costs and for
nearly identical players we study the large population limit, tending to
infinity, and find a unique quadratic-Gaussian solution of the pair of Mean
Field Game HJB-KFP equations. Examples of explicit solutions are given, in
particular for consensus problems.Comment: 31 page
A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks
In this paper we propose a LWR-like model for traffic flow on networks which
allows one to track several groups of drivers, each of them being characterized
only by their destination in the network. The path actually followed to reach
the destination is not assigned a priori, and can be chosen by the drivers
during the journey, taking decisions at junctions.
The model is then used to describe three possible behaviors of drivers,
associated to three different ways to solve the route choice problem: 1.
Drivers ignore the presence of the other vehicles; 2. Drivers react to the
current distribution of traffic, but they do not forecast what will happen at
later times; 3. Drivers take into account the current and future distribution
of vehicles. Notice that, in the latter case, we enter the field of
differential games, and, if a solution exists, it likely represents a global
equilibrium among drivers.
Numerical simulations highlight the differences between the three behaviors
and suggest the existence of multiple Wardrop equilibria
On the pointwise convergence of the integral kernels in the Feynman-Trotter formula
We study path integrals in the Trotter-type form for the Schr\"odinger
equation, where the Hamiltonian is the Weyl quantization of a real-valued
quadratic form perturbed by a potential in a class encompassing that -
considered by Albeverio and It\^o in celebrated papers - of Fourier transforms
of complex measures. Essentially, is bounded and has the regularity of a
function whose Fourier transform is in . Whereas the strong convergence in
in the Trotter formula, as well as several related issues at the operator
norm level are well understood, the original Feynman's idea concerned the
subtler and widely open problem of the pointwise convergence of the
corresponding probability amplitudes, that are the integral kernels of the
approximation operators. We prove that, for the above class of potentials, such
a convergence at the level of the integral kernels in fact occurs, uniformly on
compact subsets and for every fixed time, except for certain exceptional time
values for which the kernels are in general just distributions. Actually,
theorems are stated for potentials in several function spaces arising in
Harmonic Analysis, with corresponding convergence results. Proofs rely on
Banach algebras techniques for pseudo-differential operators acting on such
function spaces.Comment: 26 page
Nearly Optimal Patchy Feedbacks for Minimization Problems with Free Terminal Time
The paper is concerned with a general optimization problem for a nonlinear
control system, in the presence of a running cost and a terminal cost, with
free terminal time. We prove the existence of a patchy feedback whose
trajectories are all nearly optimal solutions, with pre-assigned accuracy.Comment: 13 pages, 3 figures. in v2: Fixed few misprint
Energetic stability and magnetic properties of Mn dimers in silicon
We present an accurate first-principles study of magnetism and energetics of single Mn impurities and Mn dimers in Si. Our results, in general agreement with available experiments, show that (i) Mn atoms tend to aggregate, the formation energy of dimers being lower than the sum of the separate constituents, (ii) ferromagnetic coupling is favored between the Mn atoms constituting the dimers in p-type Si, switching to an antiferromagnetic coupling in n-type Si, (iii) Mn atoms show donors (acceptor) properties in p-type (n-type) Si, therefore they tend to compensate doping, while dimers being neutral or acceptors allow for Si to be doped p-type. (C) 2004 American Institute of Physics
Gap opening in ultrathin Si layers: Role of confined and interface states
We present first principle calculations of ultrathin silicon (111) layers embedded in CaF2, a lattice matched insulator. Our all electron calculation allows a check of the quantum confinement hypothesis for the Si band gap opening as a function of thickness. We find that the gap opening is mostly due to the valence band while the lowest conduction band states shift very modestly due to their pronounced interface character. The latter states are very sensitive to the sample design. We suggest that a quasidirect band gap can be achieved by stacking Si layers of different thickness
- …