Self-Adjoint Local Boundary Problems on Compact Surfaces. I. Spectral Flow

Abstract

The paper presents a first step towards a family index theorem for classical self-adjoint boundary value problems. We address here the simplest nontrivial case of manifolds with boundary, namely the case of two-dimensional manifolds. The first result of the paper is an index theorem for families of first order self-adjoint elliptic differential operators with local boundary conditions, parametrized by points of a compact topological space X. We compute the K 1 (X)valued index in terms of the topological data over the boundary. The second result is universality of the index: we show that the index is a universal additive homotopy invariant for such families, if the vanishing on families of invertible operators is required. Contents1. ind a (γ 1 ) = ind a (γ 2 ). Family index for self-adjoint unbounded operatorsIn order to deal with unbounded self-adjoint operators (in particular, with self-adjoint differential operators) directly, one needs an analogue of the Atiyah-Singer theory [2]. Cf.[5], [7], [9]. This section is devoted to such an analogue adapted to our framework. The functor K 1 . Let H be a Hilbert space. Denote by B(H) the space of bounded linear operators H → H with the norm topology.

Self-adjoint local boundary problems on compact surfaces. II. Family index

“…Family index. In this paper, we generalize the results of [23] to families of such boundary value problems parametrized by points of an arbitrary compact space X. Our results may be viewed as a first step towards a general family index theorem for classical self-adjoint boundary value problems.…”

“…As was shown by the author in [23,Proposition 4.3], self-adjoint elliptic local boundary conditions L for A are in a one-to-one correspondence with self-adjoint bundle automorphisms T of E @ . This correspondence is given by the rule L D Ker P T with P T D P C 1 C i .n/ 1 TP ;…”

“…Topological data. Following [23], from an element .A; L/ 2 Ell.E/ we extract a topological data which contains all the information we need to compute the family index. These data are encoded in a vector subbundle F D F .A; L/ of E @ , which depends only on L and the restriction of the symbol of A to the boundary.…”

“…Unfortunately, the methods of [10,15,26] use essentially the specific nature of Dirac operators and cannot be applied to more general classes of self-adjoint elliptic differential operators. In [23], the author generalized results of [21] in a different direction, computing the spectral flow for curves of arbitrary first order self-adjoint elliptic differential operators on a compact oriented surface with boundary.…”

“…As well as in [23], we address here the simplest non-trivial case of manifolds with boundary, namely the case of two-dimensional manifolds. We consider first order selfadjoint elliptic differential operators on such manifolds, with local, or classical, boundary conditions (that is, boundary conditions defined by general pseudo-differential operators,…”