Index formulas on stratified manifolds

Elliptic operators on stratified manifolds with any finite number of strata are considered. Under certain assumptions on the symbols of operators, we obtain index formulas, which express index as a sum of indices of elliptic operators on the strata.Comment: 17 pages, no figure

New formulas for Maslov's canonical operator in a neighborhood of focal points and caustics in 2D semiclassical asymptotics

We suggest a new representation of Maslov’s canonical operator in a neighborhood of caustics using a special class of coordinate systems (eikonal coordinates) on Lagrangian manifolds. We present the results in the two-dimensional case and illustrate them with examples

Elliptic operators on manifolds with singularities and K-homology

It is well known that elliptic operators on a smooth compact manifold are classified by K-homology. We prove that a similar classification is also valid for manifolds with simplest singularities: isolated conical points and fibered boundary. The main ingredients of the proof of these results are: an analog of the Atiyah-Singer difference construction in the noncommutative case and an analog of Poincare isomorphism in K-theory for our singular manifolds. As applications we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with singularities and a formula for K-groups of algebras of pseudodifferential operators.Comment: revised version; 25 pages; section with applications expande

On Semiclassical Asymptotics for Nonlocal Equations

Abstract We consider semiclassical operators equal to linear combinations of quantized canonical transformations with pseudodifferential operators as coefficients and study semiclassical asymptotics of the solutions of the corresponding equations. Under the assumption that the group of canonical transformations is finite, we reduce our problem to a similar problem for a matrix semiclassical pseudodifferential operator. The latter problem can be treated by standard methods. DOI 10.1134/S106192082204013

Sobolev Problems with Spherical Mean Conditions and Traces of Quantized Canonical Transformations

We consider Sobolev problems (problems for an elliptic operator on a closed manifold with conditions on a closed submanifold) for the case in which these conditions are of nonlocal nature and include weighted spherical means of the unknown function over spheres of a given radius. For such problems, we establish a criterion for the Fredholm property and, in some special cases, obtain index formulas. © 2019, Pleiades Publishing, Ltd

C-Algebras of Transmission Problems and Elliptic Boundary Value Problems with Shift Operators

Abstract: We study the Fredholm solvability for a new class of nonlocal boundary value problems associated with group actions on smooth manifolds. Namely, we consider the case in which the group action is defined on an ambient manifold without boundary and does not preserve the manifold with boundary on which the problem is stated. In particular, the group action does not map the boundary into itself. The orbits of the boundary under the group action split the manifold into subdomains, and this decomposition, being combined with the C*-algebra techniques, plays an important role in our approach to the analysis of the problem. © 2022, Pleiades Publishing, Ltd