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

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