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

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 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

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

Framework and baseline examination of the German National Cohort (NAKO)

The German National Cohort (NAKO) is a multidisciplinary, population-based prospective cohort study that aims to investigate the causes of widespread diseases, identify risk factors and improve early detection and prevention of disease. Specifically, NAKO is designed to identify novel and better characterize established risk and protection factors for the development of cardiovascular diseases, cancer, diabetes, neurodegenerative and psychiatric diseases, musculoskeletal diseases, respiratory and infectious diseases in a random sample of the general population. Between 2014 and 2019, a total of 205,415 men and women aged 19–74 years were recruited and examined in 18 study centres in Germany. The baseline assessment included a face-to-face interview, self-administered questionnaires and a wide range of biomedical examinations. Biomaterials were collected from all participants including serum, EDTA plasma, buffy coats, RNA and erythrocytes, urine, saliva, nasal swabs and stool. In 56,971 participants, an intensified examination programme was implemented. Whole-body 3T magnetic resonance imaging was performed in 30,861 participants on dedicated scanners. NAKO collects follow-up information on incident diseases through a combination of active follow-up using self-report via written questionnaires at 2–3 year intervals and passive follow-up via record linkages. All study participants are invited for re-examinations at the study centres in 4–5 year intervals. Thereby, longitudinal information on changes in risk factor profiles and in vascular, cardiac, metabolic, neurocognitive, pulmonary and sensory function is collected. NAKO is a major resource for population-based epidemiology to identify new and tailored strategies for early detection, prediction, prevention and treatment of major diseases for the next 30 years. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s10654-022-00890-5