Index defects in the theory of spectral boundary value problems

Mathematische Nachrichten

Sternin

2005

Self Cite

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the C * -algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah-Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given.

“…This gives boundary value problems in subspaces (see [18]); 3) π : ∂M → X and π is a covering. This gives a class of nonlocal boundary value problems studied in [14].…”

Section: 1 Main Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Boundary value problems on manifolds with fibered boundary

Mathematische Nachrichten

Sternin

2005

Self Cite

“…The boundary condition in (5.9) relates the values of U at t = 0 and t = 1, i.e., it is a nonlocal condition. Nonlocal boundary value problems of this type were considered in [27]. Let us show that this problem is elliptic in the sense of the cited paper.…”

Section: Homotopy Of the Boundary Conditionmentioning

confidence: 99%

“…Let us show that the boundary value problem (5.11) is elliptic, i.e., it satisfies the Shapiro-Lopatinskii condition (e.g., see [25]). To prove this, we consider the Calderon bundle [25] (see also [27])…”

Section: Homotopy Of the Boundary Conditionmentioning

confidence: 99%

On the index of elliptic operators associated with a diffeomorphism of a manifold

2010

Dokl. Math.

We develop elliptic theory of operators associated with a diffeomorphism of a closed smooth manifold. The aim of the present paper is to obtain an index formula for such operators in terms of topological invariants of the manifold and of the symbol of the operator. The symbol in this situation is an element of a certain crossed product. We express the index as the pairing of the class in K-theory defined by the symbol and the Todd class in periodic cyclic cohomology of the crossed product.

“…Boundary value problems similar to Carleman's problem, with the boundary condition relating the values of the unknown function at different points on the boundary were considered (see monograph of Antonevich, Belousov and Lebedev [1] and the references cited there). Finiteness theorems were proved and for the case of finite group actions index theorems were obtained (see also [48]). On the other hand, nonlocal boundary value problems, in which the boundary condition relates the values of a function on the boundary of the domain and on submanifolds, which lie inside the domain there were considered in [15,[62][63][64].…”

mentioning

confidence: 99%

Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds

Pseudo-Differential Operators, Generalized Functions and Asymptotics

Sternin

2013

Self Cite

Abstract. In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as results obtained recently. The paper consists of an introduction and three sections. In the introduction we give a general overview of the area of research. For the reader's convenience here we tried to keep special terminology to a minimum. In the remaining sections we give detailed formulations of the most important results mentioned in the introduction.