The index of self-adjoint Shapiro-Lopatinskii boundary problems of order one

TL;DR

This paper develops an index theorem for families of self-adjoint elliptic boundary problems of order one, revealing new phenomena and obstructions unique to the self-adjoint case, with applications to generalizing existing results.

Contribution

It extends the Atiyah-Bott-Singer index theorem to self-adjoint boundary problems, identifying new obstructions and removing previous technical assumptions.

Findings

Defined a topological index for self-adjoint boundary problems.

Identified obstructions to realizing symbols as self-adjoint boundary problems.

Generalized previous results by Gorokhovsky and Lesch.

Abstract

The paper is devoted to an analogue of Atiyah-Bott-Singer index theorem for families of self-adjoint elliptic (i.e. satisfying the Shapiro-Lopatinskii condition) local boundary problems of order 1. The proofs are based on classical topological and pseudo-differential methods, but in the self-adjoint case one encounters some new phenomena. The topological index is defined following Atiyah-Bott, but in the self-adjoint case one encounters an obstruction not present in the classical situation. The analytical index is defined with the help of author's approach arXiv:2111.15081, which generalized the one of Atiyah-Singer. On the analytic index side one encounters an obstruction to the realization of symbols by self-adjoint boundary problems, similar to the obstruction to defining the topological index. As an application, we generalize results of Gorokhovsky and Lesch arXiv:1310.0210. In the first version of this paper the index theorem was proved only under an additional technical assumption. A theory of multiplicative properties of symbols and operators, developed in the second version, allows to remove this assumption.