The a priori Tan Theta Theorem for spectral subspaces

Let A be a self-adjoint operator on a separable Hilbert space H. Assume that the spectrum of A consists of two disjoint components s_0 and s_1 such that the set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint operator on H off-diagonal with respect to the partition spec(A)=s_0 \cup s_1. It is known that if ||V||<\sqrt{2}d, where d=\dist(s_0,s_1), then the perturbation V does not close the gaps between s_0 and s_1 and the spectrum of the perturbed operator L=A+V consists of two isolated components s'_0 and s'_1 grown from s_0 and s_1, respectively. Furthermore, it is known that if V satisfies the stronger bound ||V||< d then the following sharp norm estimate holds: ||E_L(s'_0)-E_A(s_0)|| \leq sin(arctan(||V||/d)), where E_A(s_0) and E_L(s'_0) are the spectral projections of A and L associated with the spectral sets s_0 and s'_0, respectively. In the present work we prove that this estimate remains valid and sharp also for d \leq ||V||< \sqrt{2}d, which completely settles the issue.Comment: v3: some typos fixed; Examples adde

Dichotomous Hamiltonians with Unbounded Entries and Solutions of Riccati Equations

An operator Riccati equation from systems theory is considered in the case that all entries of the associated Hamiltonian are unbounded. Using a certain dichotomy property of the Hamiltonian and its symmetry with respect to two different indefinite inner products, we prove the existence of nonnegative and nonpositive solutions of the Riccati equation. Moreover, conditions for the boundedness and uniqueness of these solutions are established.Comment: 31 pages, 3 figures; proof of uniqueness of solutions added; to appear in Journal of Evolution Equation

Eigenvalue estimates for non-selfadjoint Dirac operators on the real line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with non-Hermitian potential $V$ lie in the disjoint union of two disks in the right and left half plane, respectively, provided that the $L^1-norm$ of $V$ is bounded from above by the speed of light times the reduced Planck constant. An analogous result for the Schr\"odinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on $V$ implies the absence of nonreal eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials

Testbed for the Pellet Launching System for JT-60SA

As part of the European contribution to the large size superconducting tokamak project JT-60SA, a new Pellet Launching System (PLS) is designed and built. The aims of the PLS are to provide efficient fuelling to the plasma and to control and mitigate Edge Localised Modes (ELMs). Two pellet sources, one for fuelling pellets, one for pacing pellets, are delivering pellets to a centrifuge launcher. The centrifuge enables precise launch of pellets according to already proven control schemes. Furthermore, this system opens a way towards a test bed for the EU-DEMO fuelling system. The new PLS has to be completed and commissioned first at the IPP Garching pellet lab and then to be shipped to QST Naka site after having demonstrated its performance. This dedicated test bed has been set up, providing suitable vacuum conditions to operate the PLS in similar conditions (except magnetic field and radiation). Maximum hydrogen throughput is about 400 mbar·L/s per pellet source. Safety issues must be considered for hydrogen inventory of pellet sources (∼100 bar·L each). In a first step, the pellet sources will be put on a test vessel providing inherent safety by a huge volume (10 m³) which makes sure that the hydrogen concentration is below 1% under all circumstances. A hydrogen safety survey prior to assembly confirmed the concept to be followed by an assessment after the installation in order to get the required license for operation. The PLS as a whole, for the time being equipped with two pellet sources, is to be certified according to explosion prevention rules (ATEX) as a product to be shipped to Naka site. To obtain this, an appropriate declaration of explosion zones inside the vacuum system and the use of suitable and certified equipment is mandatory. Such, the integration of this system can be planned and assessed on a clear technical and regulatory basis

Spectral analysis and domain truncation for Maxwell's equations

We analyse how the spectrum of the anisotropic Maxwell system with bounded conductivity σ on a Lipschitz domain Ω is approximated by domain truncation. First we prove a new non-convex enclosure for the spectrum of the Maxwell system, with weak assumptions on the geometry of Ω and none on the behaviour of the coefficients at infinity. We also establish a simple criterion for non-accumulation of eigenvalues at as well as resolvent estimates. For asymptotically constant coefficients, we describe the essential spectrum and show that spectral pollution may occur only in the essential numerical range of the quadratic pencil , acting on divergence-free vector fields. Further, every isolated spectral point of the Maxwell system lying outside and outside the part of the essential spectrum on is approximated by spectral points of the Maxwell system on the truncated domains. Our analysis is based on two new abstract results on the (limiting) essential spectrum of polynomial pencils and triangular block operator matrices, which are of general interest. We believe our strategy of proof could be used to establish domain truncation spectral exactness for more general classes of non-self-adjoint differential operators and systems with non-constant coefficients

Dissipative Properties of Systems Composed of High-Loss and Lossless Components

We study here dissipative properties of systems composed of two components one of which is highly lossy and the other is lossless. A principal result of our studies is that all the eigenmodes of such a system split into two distinct classes characterized as high-loss and low-loss. Interestingly, this splitting is more pronounced the higher the loss of the lossy component. In addition, the real frequencies of the high-loss eigenmodes can become very small and even can vanish entirely, which is the case of overdamping.Comment: Revision; Improved exposition and typos corrected; 45 pages, 4 figure

Assessing the Criteria for Definition of Perimembranous Ventricular Septal Defects in Light of the Search for Consensus

Background: Discussions continue as to whether ventricular septal defects are best categorized according to their right ventricular geography or their borders. This is especially true when considering the perimembranous defect. Our aim, therefore, was to establish the phenotypic feature of the perimembranous defect, and to establish the ease of distinguishing its geographical variants. Methods and results: We assessed unrepaired isolated perimembranous ventricular defects from six historic archives, subcategorizing them using the ICD-11 coding system. We identified 365 defects, of which 94 (26%) were deemed to open centrally, 168 (46%) to open to the outlet, and 84 (23%) to the inlet of the right ventricle, with 19 (5%) being confluent. In all hearts, the unifying phenotypic feature was fibrous continuity between the leaflets of the mitral and tricuspid valves. This was often directly between the valves, but in all instances incorporated continuity through the atrioventricular portion of the membranous septum. In contrast, we observed fibrous continuity between the leaflets of the tricuspid and aortic valves in only 298 (82%) of the specimens. When found, discontinuity most commonly was seen in the outlet and central defects. There were no discrepancies between evaluators in distinguishing the borders, but there was occasional disagreement in determining the right ventricular geography of the defect. Conclusions: The unifying feature of perimembranous defects, rather than being aortic-to-tricuspid valvar fibrous continuity, is fibrous continuity between the leaflets of the atrioventricular valves. While right ventricular geography is important in classification, it is the borders which are more objectively defined