Nonlocal boundary value problems: index theory and semiclassical asymptotics
Nonlocal boundary value problems: index theory and semiclassical asymptotics

Nonlocal problems arise in many areas of mathematics, as well as in its applications to science and technology. In this project we focus on nonlocal boundary value problems related to the actions of groups on manifolds. The problem of studying nonlocal elliptic problems is considered. Namely, it is necessary to obtain the concept of a symbol, prove the Fredholm property of elliptic problems, obtain a homotopy classification and present index formulas for new classes of such problems.

In addition, the case of hyperbolic equations (or, more generally, equations with real characteristics) remains practically unexplored in the case of nonlocal problems, mainly due to the complexity of the subject. The task here is to extend existing and well-established methods for local problems, such as Fourier integral operators and the canonical Maslov operator, to cover nonlocal boundary value problems of the types considered here.

Project goals
  • Elliptic and hyperbolic nonlocal boundary value problems related to the actions of groups in the noninvariant case when the action does not preserve the boundary are studied. Our aim is to study both the analytical aspects of the theory (to introduce the concept of ellipticity, to prove the Fredholm property of elliptic elements), and to apply the methods of topology and noncommutative geometry to obtain index formulas. Of importance is splitting a manifold with an edge into parts by images of the boundary under diffeomorphisms given by the group action. This partition is considered as a manifold with singularities, and we are going to apply singular analysis methods to our problem. To study hyperbolic nonlocal boundary value problems, we will use semiclassical methods.
Project leader All participants
Anton Savin

Anton Savin

Doctor of Sciences, Professor, the leader of the project
Project results
Trajectory symbols are defined for nonlocal boundary value problems and problems with conjugation conditions corresponding to infinite groups of transformations that do not preserve a submanifold with boundary. The Fredholm criterion of such problems is established.
A new approach to finding formulas for the index of boundary value problems based on the theory of cyclic cohomology is given. As an example, cyclic cocycles on the algebra of symbols in the two-dimensional case are defined. The index problem for elliptic operators associated with shifts of an infinite cylinder is investigated. The index formula is presented in terms of the main symbol of such operators.
A general construction of semiclassical asymptotics for nonlocal (pseudo) differential operators with a small parameter by derivatives associated with the action of the diffeomorphism group of a manifold or canonical transformations of a cotangent bundle is developed. A discrete version of nonlocal operators – operators on a lattice is studied. Applications are obtained to the problem of propagation of wave packets on a homogeneous tree and to the Feynman checkers model of electron motion on a one-dimensional lattice.
Application area
  • The results will be used to find semiclassical asymptotics of solutions to nonlocal hyperbolic problems. They are important both in the theory of partial differential equations as well as in noncommutative geometry and global analysis. The obtained results can also be applied in the study of nonlocal boundary value problems arising in mechanics.