Layer potentials C*-algebras of domains with conical points

Carvalho, Catarina; Qiao, Yu

doi:10.2478/s11533-012-0066-y

Cited by 11 publications

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“…the fibers of the domain map d are connected). There are analysis problems, however, when one is lead to nond-connected groupoids [25].…”

Section: 5mentioning

confidence: 99%

Analysis on singular spaces: Lie manifolds and operator algebras

Nistor

2016

Journal of Geometry and Physics

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We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds--called "Lie manifolds"--that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here--work that spans over close to two decades--was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.Comment: 43 pages, based on my four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 201

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“…the fibers of the domain map d are connected). There are analysis problems, however, when one is lead to nond-connected groupoids [25].…”

Section: 5mentioning

confidence: 99%

Analysis on singular spaces: Lie manifolds and operator algebras

Nistor

2016

Journal of Geometry and Physics

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“…As mentioned, the L p -theory for infinite straight cones and wedges was considered in [19]. Also for the infinite straight cone, the generalized eigensolutions to the transmission problem (3) were explicitly computed in [32,41,48] -these will be important in our determination of the spectrum of K Γ : E → E. For the more general type of infinite cone Γ = R + ω, where ω is a smooth curve on the sphere, the invertibility of K Γ − z on certain weighted Sobolev spaces has via Mellin convolutions been reduced to the invertibility of a parametric system of operators on ω [9,45].…”

Section: Introductionmentioning

confidence: 99%

The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points

Helsing

Perfekt

2018

Journal de Mathématiques Pures et Appliquées

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We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann-Poincaré operator), as a map on the boundary surface Γ of a domain in R 3 with conical points. The spectrum of this operator directly reflects the well-posedness of related transmission problems across Γ. In particular, if the domain is understood as an inclusion with complex permittivity , embedded in a background medium with unit permittivity, then the polarizability tensor of the domain is well-defined when ( + 1)/( − 1) belongs to the resolvent set in energy norm. We study surfaces Γ that have a finite number of conical points featuring rotational symmetry. On the energy space, we show that the essential spectrum consists of an interval. On L 2 (Γ), i.e. for square-integrable boundary data, we show that the essential spectrum consists of a countable union of curves, outside of which the Fredholm index can be computed as a winding number with respect to the essential spectrum. We provide explicit formulas, depending on the opening angles of the conical points. We reinforce our study with very precise numerical experiments, computing the energy space spectrum and the spectral measures of the polarizability tensor in two different examples. Our results indicate that the densities of the spectral measures may approach zero extremely rapidly in the continuous part of the energy space spectrum. RésuméNousétudions l'adjoint du potentiel de double couche associéà l'opérateur de Laplace (l'adjoint de l'opérateur de Neumann-Poincaré) défini sur la frontière Γ d'un domaine de R 3 contenant des points coniques. Le spectre de cet opérateur est intimement liéà la résolution de problèmes de transmissionà travers Γ. En particulier, dans le contexte de la propagation des ondesélectromagnétiques, si le domaine délimité par Γ représente une inclusion contenant un matériau de permittivité complexe , immergé dans un milieu infini de permittivitéégaleà 1, on peut définir le tenseur de polarisabilité dès que le rapport ( + 1)/( − 1) appartientà l'ensemble résolvent de l'opérateur au sens de la norme d'énergie. Nousétudions des surfaces Γ qui possèdent un nombre fini de points coniquesà symétrie de rotation. Lorsque l'opérateur est défini sur l'espace d'énergie, nous montrons que son spectre essentiel est un intervalle. Lorsqu'il est défini dans l'espace L 2 (Γ), i.e. pour des fonctions de carré intégrable sur Γ, nous montrons que son spectre est constitué d'une union de courbes, en dehors desquelles on peut calculer l'indice de Fredholm de l'opérateur, comme l'indice par rapportà ces courbes. Nous donnons des formules explicites, en fonction de l'angle d'ouverture des points coniques. Nous complétons notreétude par des expériences numériques très précises, où, pour deux exemples, nous calculons le spectre de l'opérateur au sens de l'espace d'énergie et les mesures spectrales du tenseur de polarisabilité. Nos résultats suggèrent que les densités des mesures spectrales approchent zéro extrêmement rapidement dans la partie cont...

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“…The resulting differential operators Diff(V sc ) and the associated pseudodifferential ooperators are the SG-operators of [19,74,88,90,89] (called "scattering operators" in [59]). They can be obtained by considering the groupoid (14) G…”

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confidence: 99%

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

Carvalho

Nistor

Qiao

2018

Operator Theory, Operator Algebras, and Matrix Theory

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We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators, in a sense to be made precise, are invertible. Central to our theoretical results is the concept of a Fredholm groupoid, which is the class of groupoids for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel's property as the groupoids for which the regular representations are exhaustive. We show that the class of "stratified submersion groupoids" has Exel's property, where stratified submersion groupoids are defined by glueing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is explored to yield Fredholm conditions not only in the above mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.

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Layer potentials C*-algebras of domains with conical points

Cited by 11 publications

References 50 publications

Analysis on singular spaces: Lie manifolds and operator algebras

Analysis on singular spaces: Lie manifolds and operator algebras

The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

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