The functional model for maximal dissipative operators (translation form): An approach in the spirit of operator knots

In this article we develop a functional model for a general maximal dissipative operator. We construct the selfadjoint dilation of such operators. Unlike previous functional models, our model is given explicitly in terms of parameters of the original operator, making it more useful in concrete applications. For our construction we introduce an abstract framework for working with a maximal dissipative operator and its anti-dissipative adjoint and make use of the ˇStraus characteristic function in our setting. Explicit formulae are given for the selfadjoint dilation, its resolvent, a core and the completely non-selfadjoint subspace; minimality of the dilation is shown. The abstract theory is illustrated by the example of a Schrödinger operator on a half-line with dissipative potential, and boundary condition and connections to existing theory are discussed

Conditions for the spectrum associated with a leaky wire to contain the interval [− α2/4, ∞)

The method of singular sequences is used to provide a simple and, in some respects, a more general proof of a known spectral result for leaky wires. The method introduces a new concept of asymptotic straightness

Inverse problems for boundary triples with applications

This paper discusses the inverse problem of how much information on an operator can be determined/detected from `measurements on the boundary'. Our focus is on non-selfadjoint operators and their detectable subspaces (these determine the part of the operator `visible' from `boundary measurements'). We show results in an abstract setting, where we consider the relation between the M-function (the abstract Dirichlet to Neumann map or the transfer matrix in system theory) and the resolvent bordered by projections onto the detectable subspaces. More specifically, we investigate questions of unique determination, reconstruction, analytic continuation and jumps across the essential spectrum. The abstract results are illustrated by examples of Schr?odinger operators, matrix differential operators and, mostly, by multiplication operators perturbed by integral operators(the Friedrichs model), where we use a result of Widom to show that the detectable subspace can be characterized in terms of an eigenspace of a Hankel-like operator

Weyl Solutions and jj-selfadjointness for Dirac Operators

We consider a non-selfadjoint Dirac-type differential expression $$(0.1)D(Q)y := J_n {dy \over dx} +Q(x)y$$ with a non-selfadjoint potential matrix $Q$ $\epsilon$ $L$ ^{1}_{loc}(\scr J, \Bbb C^{n \times n}) and a signature matrix $J_n=-J^{-J}_n=-J^*_n$ $\epsilon$ $\Bbb C ^{n \times n}$. Here \scr J denotes either the line $\Bbb R$ or the half-line $\Bbb R_+$. With this differential expression one associates in L^2(\scr J, \Bbb C^n) the (closed) maximal and minimal operators $D_{max}(Q)$ and $D_{min}(Q)$, respectively. One of our main results for the whole line case states that $D_{max}(Q)=D_{min}(Q)$ in $L^2$ $\Bbb R, \Bbb C^n$. Moreover, we show that if the minimal operator $D_{min}(Q)$ in $L^2(\Bbb R, \Bbb C^n)$ is $j$-symmetric with respect to an appropriate involution $j$, then it is $j$-selfadjoint. Similar results are valid in the case of the semiaxis $\Bbb R_+$. In particular, we show that if $n=2p$ and the minimal operator $D^+_{min}(Q)$ in $L^2(\Bbb R_+,\Bbb C^{2p})$ is (\j\)-symmetric, then there exists a $2p \times p-$Weyl-type matrix solution. $\Psi(z,\cdot)$ $\epsilon$ $L^2(\Bbb R_+, \Bbb C^{2p \times p})$ of the equation $D^+_{max}(Q)\Psi(z,\cdot)=z \Psi(z,\cdot)$. A similar result is valid for the expression (0.1) whenever there exists a proper extension $\tilde A$ with dim (dom $\tilde A$/dom $D^+_{min}(Q))=p$ and nonempty resolvent set. In particular, it holds if a potential matrix (\Q\) has a bounded imaginary part. This leads to the existence of a unique Weyl function for the express (0.1). The main results are proven by means of a reduction to the self-adjoint case by using the technique of dual pairs of operators. The differential expression (0.1) is of significance as it appears in the Lax formulation of the vector valued nonlinear Schrödinger equation

The inverse problem for a spectral asymmetry function of the Schrodinger operator on a finite interval

For the Schroedinger equation $−d^2u/dx^2 + q(x)u = λu$ on a finite $x$-interval, there is defined an “asymmetry function” $a(λ; q)$, which is entire of order 1/2 and type 1 in $λ$. Our main result identifies the classes of square-integrable potentials $q(x)$ that possess a common asymmetry function $a(λ)$. For any given $a(λ)$, there is one potential for each Dirichlet spectral sequence

Essential Spectrum for Maxwell’s Equations

We study the essential spectrum of operator pencils associated with anisotropic Maxwell equations, with permittivity ε, permeability μ and conductivity σ, on finitely connected unbounded domains. The main result is that the essential spectrum of the Maxwell pencil is the union of two sets: namely, the spectrum of the pencil div((ωε+iσ)∇⋅), and the essential spectrum of the Maxwell pencil with constant coefficients. We expect the analysis to be of more general interest and to open avenues to investigation of other questions concerning Maxwell’s and related systems

Effects of antiplatelet therapy on stroke risk by brain imaging features of intracerebral haemorrhage and cerebral small vessel diseases: subgroup analyses of the RESTART randomised, open-label trial

Background Findings from the RESTART trial suggest that starting antiplatelet therapy might reduce the risk of recurrent symptomatic intracerebral haemorrhage compared with avoiding antiplatelet therapy. Brain imaging features of intracerebral haemorrhage and cerebral small vessel diseases (such as cerebral microbleeds) are associated with greater risks of recurrent intracerebral haemorrhage. We did subgroup analyses of the RESTART trial to explore whether these brain imaging features modify the effects of antiplatelet therapy

Antimicrobial resistance among migrants in Europe: a systematic review and meta-analysis

BACKGROUND: Rates of antimicrobial resistance (AMR) are rising globally and there is concern that increased migration is contributing to the burden of antibiotic resistance in Europe. However, the effect of migration on the burden of AMR in Europe has not yet been comprehensively examined. Therefore, we did a systematic review and meta-analysis to identify and synthesise data for AMR carriage or infection in migrants to Europe to examine differences in patterns of AMR across migrant groups and in different settings. METHODS: For this systematic review and meta-analysis, we searched MEDLINE, Embase, PubMed, and Scopus with no language restrictions from Jan 1, 2000, to Jan 18, 2017, for primary data from observational studies reporting antibacterial resistance in common bacterial pathogens among migrants to 21 European Union-15 and European Economic Area countries. To be eligible for inclusion, studies had to report data on carriage or infection with laboratory-confirmed antibiotic-resistant organisms in migrant populations. We extracted data from eligible studies and assessed quality using piloted, standardised forms. We did not examine drug resistance in tuberculosis and excluded articles solely reporting on this parameter. We also excluded articles in which migrant status was determined by ethnicity, country of birth of participants' parents, or was not defined, and articles in which data were not disaggregated by migrant status. Outcomes were carriage of or infection with antibiotic-resistant organisms. We used random-effects models to calculate the pooled prevalence of each outcome. The study protocol is registered with PROSPERO, number CRD42016043681. FINDINGS: We identified 2274 articles, of which 23 observational studies reporting on antibiotic resistance in 2319 migrants were included. The pooled prevalence of any AMR carriage or AMR infection in migrants was 25·4% (95% CI 19·1-31·8; I2 =98%), including meticillin-resistant Staphylococcus aureus (7·8%, 4·8-10·7; I2 =92%) and antibiotic-resistant Gram-negative bacteria (27·2%, 17·6-36·8; I2 =94%). The pooled prevalence of any AMR carriage or infection was higher in refugees and asylum seekers (33·0%, 18·3-47·6; I2 =98%) than in other migrant groups (6·6%, 1·8-11·3; I2 =92%). The pooled prevalence of antibiotic-resistant organisms was slightly higher in high-migrant community settings (33·1%, 11·1-55·1; I2 =96%) than in migrants in hospitals (24·3%, 16·1-32·6; I2 =98%). We did not find evidence of high rates of transmission of AMR from migrant to host populations. INTERPRETATION: Migrants are exposed to conditions favouring the emergence of drug resistance during transit and in host countries in Europe. Increased antibiotic resistance among refugees and asylum seekers and in high-migrant community settings (such as refugee camps and detention facilities) highlights the need for improved living conditions, access to health care, and initiatives to facilitate detection of and appropriate high-quality treatment for antibiotic-resistant infections during transit and in host countries. Protocols for the prevention and control of infection and for antibiotic surveillance need to be integrated in all aspects of health care, which should be accessible for all migrant groups, and should target determinants of AMR before, during, and after migration. FUNDING: UK National Institute for Health Research Imperial Biomedical Research Centre, Imperial College Healthcare Charity, the Wellcome Trust, and UK National Institute for Health Research Health Protection Research Unit in Healthcare-associated Infections and Antimictobial Resistance at Imperial College London