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

Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map

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

Ends and means: experts debate the democratic oversight of the UK’s intelligence services

Revelations from Edward Snowden about the scope of intelligence activities in the UK have led to renewed attempts to enhance democratic oversight of the UK’s security services. The heads of MI5, MI6 and GCHQ appeared before the Intelligence and Security Committee for the first time, while Lord Macdonald called for strengthened parliamentary accountability. In this post, we ask democracy and security experts to consider the need for further reform

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

Evidence that breast cancer risk at the 2q35 locus is mediated through IGFBP5 regulation.

GWAS have identified a breast cancer susceptibility locus on 2q35. Here we report the fine mapping of this locus using data from 101,943 subjects from 50 case-control studies. We genotype 276 SNPs using the 'iCOGS' genotyping array and impute genotypes for a further 1,284 using 1000 Genomes Project data. All but two, strongly correlated SNPs (rs4442975 G/T and rs6721996 G/A) are excluded as candidate causal variants at odds against >100:1. The best functional candidate, rs4442975, is associated with oestrogen receptor positive (ER+) disease with an odds ratio (OR) in Europeans of 0.85 (95% confidence interval=0.84-0.87; P=1.7 × 10(-43)) per t-allele. This SNP flanks a transcriptional enhancer that physically interacts with the promoter of IGFBP5 (encoding insulin-like growth factor-binding protein 5) and displays allele-specific gene expression, FOXA1 binding and chromatin looping. Evidence suggests that the g-allele confers increased breast cancer susceptibility through relative downregulation of IGFBP5, a gene with known roles in breast cell biology