ANALYTIC DEFORMATIONS OF THE SPECTRUM OF A FAMILY OF DIRAC OPERATORS ON AN ODD-DIMENSIONAL MANIFOLD WITH BOUNDARY - INTRODUCTION

In this paper we study the perturbation theory for the spectrum of an analytic family of formally self-adjoint Dirac operators on a manifold with boundary. Let D : C-infinity(E) --> C-infinity(E) be a Dirac ope rator on a vector bundle E over a compact oriented Riemannian manifold X with boundary, and let A be the tangential operator over delta X. W e consider the following two ways of obtaining a spectrum from the ope rator D. Both of the following require the choice of a Lagrangian subs pace L of the symplectic space ker A. The first type of spectrum, whic h we call the ''Atiyah-Patodi-Singer (APS) spectrum'' of D, is simply the spectrum of D\{phi epsilon Gamma(E) : phi/delta X epsilon L + P-+} , where P-+ denotes the span of those eigenvectors of A whose eigenval ues are positive. With these global boundary conditions, D becomes a s elf-adjoint Fredholm operator; of course the spectrum depends on one's choice of L. To obtain the second type of spectrum, which we call the ''extended L(2)'' spectrum of D, one first attaches an infinite colla r delta X x [0, infinity) to the boundary of X to form an open manifol d X(infinity), and extends the bundle E trivially along the collar. An ''extended L(2)'' eigenvector of D is a section phi over X(infinity) such that (1) D phi = lambda phi, (2) the L(2)-norm of phi\delta X x { r} is uniformly bounded with respect to r, and (3) the L(2)-projection of phi/delta X x {0} to ker A is in L. We call the set (with multipli city) of those values of X corresponding to these extended L(2) eigenv ectors the ''extended L(2) spectrum'' of D. It is easy to see that for an operator D as above, the eigenvalue 0 has the same multiplicity wi th respect to either of these types of spectrum. Our main result is th at the perturbation theories of these two types of spectrum have the f ollowing important similarity: if D-t is an analytic path of such oper ators then there is a 1-1 correspondence between those extended L(2) e igenvalues which are passing through 0 at time t = 0 and those Atiyah- Patodi-Singer eigenvalues which are passing through 0 at t = 0 and tha t this correspondence may be taken to preserve the order and sign of t he first non-vanishing derivative of each of these eigenvalues. Our ma in motivation for comparing these two types of spectrum is that we wan t to understand the perturbation theory of the APS spectrum, but we ha ve devised techniques to compute the perturbation theory of the extend ed L(2) spectrum in certain important cases. Thus the results of this paper show that our techniques are actually giving us information abou t the APS spectrum as well. To be more precise, we wish to develop coh omological methods for computing the spectral flow of the path of sign ature operators associated to a path of flat connections on a manifold with boundary analogous to those developed in [KK3] and [FL] for clos ed manifolds. For applications, it is most useful to compute the spect ral flow of the APS spectrum. However it turns out that the cohomologi cal methods we have devised give the first and, if the first vanishes, the second derivatives of the extended L(2) eigenvalues. Thus the mot ivation for the present work was to show that we could deduce from thi s the corresponding information about the APS eigenvalues. We emphasiz e that we do not assume the tangential operator is invertible along th e path, but only that the dimension of its kernel remains constant.