Rigorous numerics for NLS: bound states, spectra, and controllability

In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schr\"odinger equation (NLS); specifically, to determining bound--state solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof [6] of the local exact controllability of NLS.Comment: 30 pages, 2 figure

Regulation of genes involved in carnitine homeostasis by PPARa across different species (rat, mouse, pig, cattle, chicken, and human)

Recent studies in rodents convincingly demonstrated that PPAR-alpha is a key regulator of genes involved in carnitine homeostasis, which serves as a reasonable explanation for the phenomenon that energy deprivation and fibrate treatment, both of which cause activation of hepatic PPAR-alpha, causes a strong increase of hepatic carnitine concentration in rats. The present paper aimed to comprehensively analyse available data from genetic and animal studies with mice, rats, pigs, cows, and laying hens and from human studies in order to compare the regulation of genes involved in carnitine homeostasis by PPAR-alpha across different species. Overall, our comparative analysis indicates that the role of PPAR-alpha as a regulator of carnitine homeostasis is well conserved across different species. However, despite demonstrating a well-conserved role of PPAR-alpha as a key regulator of carnitine homeostasis in general, our comprehensive analysis shows that this assumption particularly applies to the regulation by PPAR-alpha of carnitine uptake which is obviously highly conserved across species, whereas regulation by PPAR-alpha of carnitine biosynthesis appears less well conserved across species

Local exact controllability of a 1D Bose-Einstein condensate in a time-varying box

We consider a one dimensional Bose-Einstein condensate in a in finite square-well (box) potential. This is a nonlinear control system in which the state is the wave function of the Bose Einstein condensate and the control is the length of the box. We prove that local exact controllability around the ground state (associated with a fi xed length of the box) holds generically with respect to the chemical potential ; i.e. up to an at most countable set of values. The proof relies on the linearization principle and the inverse mapping theorem, as well as ideas from analytic perturbation theory

Minimal time for the bilinear control of Schrödinger equations

International audienceWe consider a quantum particle in a potential V (x) (x in R^N) subject to a (spatially homogeneous) time-dependent electric field E(t), which plays the role of the control. Under generic assumptions on V , this system is approximately controllable on the L2(R^N;C)-sphere, in su ffiently large times T, as proved by Boscain, Caponigro, Chambrion and Sigalotti. In the present article, we show that this approximate controllability result is false in small time. As a consequence, the result by Boscain et al. is, in some sense, optimal with respect to the control time T

Minimal time for the approximate bilinear control of Schrödinger equations

International audienceWe consider a quantum particle in a potential $V (x) (x ∈ R^N)$ in a time-dependent electric eld E(t) (the control). Boscain, Caponigro, Chambrion and Sigalotti proved in [2] that, under generic assumptions on V , this system is approximately controllable on the $L^2 (R^N$ , C)-sphere, in suciently large time T. In the present article we show that approximate controllability does not hold in arbitrarily small time, no matter what the initial state is. This generalizes our previous result for Gaussian initial conditions. Moreover, we prove that the minimal time can in fact be arbitrarily large