By P. Kirk

The topic of this memoir is the spectrum of a Dirac-type operator on an odd-dimensional manifold M with boundary and, fairly, how this spectrum varies lower than an analytic perturbation of the operator. forms of eigenfunctions are thought of: first, these gratifying the "global boundary stipulations" of Atiyah, Patodi, and Singer and moment, these which expand to $L^2$ eigenfunctions on M with an unlimited collar hooked up to its boundary.

The unifying inspiration in the back of the research of those kinds of spectra is the inspiration of yes "eigenvalue-Lagrangians" within the symplectic house $L^2(\partial M)$, an concept because of Mrowka and Nicolaescu. via learning the dynamics of those Lagrangians, the authors may be able to identify that these parts of the 2 forms of spectra which go through 0 behave in primarily an identical approach (to first non-vanishing order). at times, this results in topological algorithms for computing spectral stream.

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**Example text**

7 below) using the second property (ii) shows that | | ( / d - * ( A , t ) ) t , | | ? < C 1 + C a |A|||t;|| a for some positive constants C» independent of v. Thus Id — $(\,t) is continuous as a map L2(E) —• L2(E) and so 3>(A, t) is a compact perturbation of the identity. • If V C L2(E) is a subspace, we denote by C°°(E) V) those sections u whose restriction to the boundary r(u) G L2(E) lies in V. The global boundary conditions of Atiyah-Patodi-Singer in [APS1] correspond to V = H®PQ '. Our assumptions on D together with the results of [APS1] imply that the restriction of D with Atiyah-Patodi-Singer boundary conditions D : C°°(E; « 0 P O + ) ~* C°°(E) is Fredholm, its adjoint is D:Coo(E;P0+)-^Coo(E), and its index is equal to ^dimW- Unique continuation of solutions to Dirac operators implies that the kernel of D : C°°(E; H (&PQ) —• C°°(E) is isomorphic to iVo fl (W0JPO"), a n d the kernel of its adjoint is isomorphic to NQ D PQ .

N (e-R^L 0 P+) ^ 0}. This shows how to define E^ in terms of X(R) for any R. (X(oo);E) to consist of those sections u whose restriction i(u) to the slice Y x {R} satisfy pro)ni(uj) e L. Then E ^ consists of those small A so that the extended L2 A-eigenspace of D intersects C<^)(X(oo);E) non-trivially. This last description justifies the terminology. Intuitively, one can think of this as the set of extended L2 eigenvectors whose harmonic parts pass through the "polarizing filter" L at Y x {R}. 28 ANALYTIC DEFORMATIONS 29 C o m m e n t .

Several difficulties arise when A equals an eigenvalue /i^ ^ 0. 7) the terms involving fik grow linearly in u if A = /ifc. Notice furthermore that P^ is not isotropic if |A| > /xn+i- P. KIRK E. KLASSEN 22 We remark that Px is not orthogonal to Px if A ^ 0. In fact, in the two dimensional space Sk, the lines S£x and S^x have reciprocal slopes; they are perpendicular if and only if A=0. Notice that as k —• oo, /i& —y oo, and so the angle between S£x and Sj^x approaches n/2. Consider now an analytic 1-parameter family D(t),t G (—e, e) of Dirac operators with the same principal symbol in cylindrical form, as before.