An explicit model for the adiabatic evolution of quantum observables driven by 1D shape resonances

This paper is concerned with a linearized version of the transport problem where the Schr\"{o}dinger-Poisson operator is replaced by a non-autonomous Hamiltonian, slowly varying in time. We consider an explicitly solvable model where a semiclassical island is described by a flat potential barrier, while a time dependent 'delta' interaction is used as a model for a single quantum well. Introducing, in addition to the complex deformation, a further modification formed by artificial interface conditions, we give a reduced equation for the adiabatic evolution of the sheet density of charges accumulating around the interaction point.Comment: latex; 26 page

Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator

Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator $H_\epsilon = -\partial_x^2 + x^2 + i\epsilon^{-1}f(x)$ on $L^2(R)$, where $f$ is a real-valued function and $\epsilon > 0$ a small parameter. We define $\Sigma(\epsilon)$ as the infimum of the real part of the spectrum of $H_\epsilon$, and $\Psi(\epsilon)^{-1}$ as the supremum of the norm of the resolvent of $H_\epsilon$ along the imaginary axis. Under appropriate conditions on $f$, we show that both quantities $\Sigma(\epsilon)$, $\Psi(\epsilon)$ go to infinity as $\epsilon \to 0$, and we give precise estimates of the growth rate of $\Psi(\epsilon)$. We also provide an example where $\Sigma(\epsilon)$ is much larger than $\Psi(\epsilon)$ if $\epsilon$ is small. Our main results are established using variational "hypocoercive" methods, localization techniques and semiclassical subelliptic estimates.Comment: 38 pages, 4 figure

Accurate WKB Approximation for a 1D Problem with Low Regularity

2000 Mathematics Subject Classification: 34L40, 65L10, 65Z05, 81Q20.This article is concerned with the analysis of the WKB expansion in a classically forbidden region for a one dimensional boundary value Schrodinger equation with a non smooth potential. The assumed regularity of the potential is the one coming from a non linear problem and seems to be the critical one for which a good exponential decay estimate can be proved for the first remainder term. The treatment of the boundary conditions brings also some interesting subtleties which require a careful application of Carleman’s method

Mean field limit for bosons and propagation of Wigner measures

We consider the N-body Schr\"{o}dinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work \cite{AmNi}, the mean field limit is translated into a semiclassical problem with a small parameter $\epsilon\to 0$, after introducing an $\epsilon$-dependent bosonic quantization. The limit is expressed as a push-forward by a nonlinear flow (e.g. Hartree) of the associated Wigner measures. These object and their basic properties were introduced in \cite{AmNi} in the infinite dimensional setting. The additional result presented here states that the transport by the nonlinear flow holds for rather general class of quantum states in their mean field limit.Comment: 10 page

Far from equilibrium steady states of 1D-Schrödinger-Poisson systems with quantum wells I

We describe the asymptotic of the steady states of the out-of equilibrium Schrödinger-Poisson system, in the regime of quantum wells in a semiclassical island. After establishing uniform estimates on the nonlinearity, we show that the nonlinear steady states lie asymptotically in a finite-dimensional subspace of functions and that the involved spectral quantities are reduced to a finite number of so-called asymptotic resonant energies. The asymptotic finite dimensional nonlinear system is written in a general setting with only a partial information on its coefficients. After this first part, a complete derivation of the asymptotic nonlinear system will be done for some specific cases in a forthcoming article. UNE VERSION MODIFIEE DE CE TEXTE EST PARUE DANS LES ANNALES DE L'INSTITUT H. POINCARE, ANALYSE NON LINEAIRE

Dynamical phase transition for a quantum particle source

We analyze the time evolution describing a quantum source for noninteracting particles, either bosons or fermions. The growth behaviour of the particle number (trace of the density matrix) is investigated, leading to spectral criteria for sublinear or linear growth in the fermionic case, but also establishing the possibility of exponential growth for bosons. We further study the local convergence of the density matrix in the long time limit and prove the semiclassical limit.Comment: 24 pages; In the new version, we added several references concerning open quantum systems and present an extended result on linear particle production in the fermionic cas

Densely Entangled Financial Systems

In [1] Zawadoski introduces a banking network model in which the asset and counter-party risks are treated separately and the banks hedge their assets risks by appropriate OTC contracts. In his model, each bank has only two counter-party neighbors, a bank fails due to the counter-party risk only if at least one of its two neighbors default, and such a counter-party risk is a low probability event. Informally, the author shows that the banks will hedge their asset risks by appropriate OTC contracts, and, though it may be socially optimal to insure against counter-party risk, in equilibrium banks will {\em not} choose to insure this low probability event. In this paper, we consider the above model for more general network topologies, namely when each node has exactly 2r counter-party neighbors for some integer r>0. We extend the analysis of [1] to show that as the number of counter-party neighbors increase the probability of counter-party risk also increases, and in particular the socially optimal solution becomes privately sustainable when each bank hedges its risk to at least n/2 banks, where n is the number of banks in the network, i.e., when 2r is at least n/2, banks not only hedge their asset risk but also hedge its counter-party risk.Comment: to appear in Network Models in Economics and Finance, V. Kalyagin, P. M. Pardalos and T. M. Rassias (editors), Springer Optimization and Its Applications series, Springer, 201

Clearing algorithms and network centrality

I show that the solution of a standard clearing model commonly used in contagion analyses for financial systems can be expressed as a specific form of a generalized Katz centrality measure under conditions that correspond to a system-wide shock. This result provides a formal explanation for earlier empirical results which showed that Katz-type centrality measures are closely related to contagiousness. It also allows assessing the assumptions that one is making when using such centrality measures as systemic risk indicators. I conclude that these assumptions should be considered too strong and that, from a theoretical perspective, clearing models should be given preference over centrality measures in systemic risk analyses

Simulation of resonant tunneling heterostructures: numerical comparison of a complete Schr{ö}dinger-Poisson system and a reduced nonlinear model

Two different models are compared for the simulation of the transverse electronic transport through an heterostructure: a $1D$ self-consistent Schr{ö}dinger-Poisson model with a numerically heavy treatment of resonant states and a reduced model derived from an accurate asymptotic nonlinear analysis. After checking the agreement at the qualitative and quantitative level on quite well understood bifurcation diagrams, the reduced model is used to tune double well configurations for which nonlinearly interacting resonant states actually occur in the complete self-consistent model

Linear vs. nonlinear effects for nonlinear Schrodinger equations with potential

We review some recent results on nonlinear Schrodinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite precisely. This includes semi-classical regimes, as well as finite time blow-up and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs.Comment: 20 pages; survey of previous result