Analysis of geometric operators on open manifolds: A groupoid approach

By Robert Lauter, Victor Nistor. 2000

87 citations recorded by the source corpus.

The first five sections of this paper are a survey of algebras of pseudodifferential operators on groupoids. We thus review differentiable groupoids, the definition of pseudodifferential operators on groupoids, and some of their properties. We use then this background material to establish a few new results on these algebras, results that are useful for the analysis of geometric operators on non-compact manifolds and singular spaces. The first step is to establish that the geometric operators on groupoids are in our algebras. This then leads to criteria for the Fredholmness of geometric operators on suitable non-compact manifolds, as well as to an inductive procedure to study their essential spectra. As an application, we answer a question of Melrose on the essential spectrum of the Laplace operator on manifolds with multi-cylindrical ends.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b66c6b311d7797cb48cc862ce26b75d6293abc3f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Victor Nistor
2. M. Crainic
3. Robert Lauter
4. Rui Loja Fernandes
5. M. Gualtieri
6. Songhao Li

In Geometry and Topology

1. Victor Nistor
2. M. Crainic
3. Rui Loja Fernandes
4. Robert Lauter
5. M. Gualtieri
6. Songhao Li

In Mathematical Physics

1. Hengguang Li
2. Victor Nistor
3. M. Crainic
4. Robert Lauter
5. Rui Loja Fernandes
6. M. Gualtieri
7. Songhao Li

Citations in the corpus

• Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM (2009, background): “See [38,58] and the references therein for the case of interior singularities due to the potential V , where no boundary is involved (in which case, a suitable pseudodifferential calculus immediately gives the desired Fredholm condition).”
• ANALYSIS OF THE FINITE ELEMENT METHOD FOR TRANSMISSION/MIXED BOUNDARY VALUE PROBLEMS ON GENERAL POLYGONAL DOMAINS ∗ (2010, unknown): “The Fredholm criterion is well known [43, 45, 73].”
• Interface and mixed boundary value problems on $n$-dimensional polyhedral domains (2010, unknown): “Documenta Mathematica 15 (2010) 687–745”
• Pseudodifferential analysis on continuous family groupoids (2000, unknown): “In this context, let us mention only the edge-calculus on manifolds with boundary [13, 16], and the b- resp.”