By P. Kirk

ISBN-10: 082180538X

ISBN-13: 9780821805381

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 less than an analytic perturbation of the operator. sorts of eigenfunctions are thought of: first, these fulfilling 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 suggestion at the back of the research of those sorts of spectra is the idea of definite "eigenvalue-Lagrangians" within the symplectic area $L^2(\partial M)$, an concept as a result of Mrowka and Nicolaescu. by way of learning the dynamics of those Lagrangians, the authors may be able to identify that these parts of the 2 different types of spectra which go through 0 behave in primarily an analogous means (to first non-vanishing order). often times, this results in topological algorithms for computing spectral circulate.

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**Additional info for Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold With Boundary**

**Sample text**

Note that the subspaces L 0 P^ and P^ 0 L are Lagrangians in L2(E) if L C ker A is a Lagrangian and A is smaller than /x n +i. We call these subspaces tangential Lagrangians, since they depend only on the tangential operator. We denote by £ those Lagrangians in L2(E) commensurate to J(L(0)+PQ~(0)) (or equivalently commensurate to P0~(0)). 4 shows that the N\(t) are in £, as well as Lagrangians of the form P£(t)®L(t) and J(L(t)®Pf(t)) for any A, t. 6 PROPOSITION. The eigenvalue Lagrangians and the negative tangential Lagrangians are contained in the space £ of Lagrangians commensurate to Po~(0).

Moreover, B is the intersection of all the V(L(t)) over all choices of L(t). The results of Chapter 4 roughly say that the section N is transverse to V(L(t)). Note that V(L(t)) has (real) codimension 1 in each fiber. On the other hand, the codimension of B goes up as the dimension of the kernel of the tangential operator goes up, so that "generically" one would would expect no ANALYTIC DEFORMATIONS 43 intersections with JB, that is, all eigenvalues are type 3. g. operators coupled to flat connections) one cannot perturb away the intersections with B; the discontinuities of the reduction r : U x C —> £ cannot be avoided.

2 can be used to show that near zero, the real-analytic subset of C consisting of those A which are eigenvalues of D acting on L2,~6 is a real analytic set transverse to the real axis. CHAPTER 5 D Y N A M I C PROPERTIES OF T H E EIGENVALUE L A G R A N G I A N S NR AS R -+ oo We next explain what happens to NR as R —> oo. This gives information about the behavior of eigenvalues of D : C°°(E; L 0 P0+) -> C°°(£) on X ( P ) as P —• oo, since A is an eigenvalue of D on X(R) if and only if NR n L 0 P ^ ^ 0.

### Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold With Boundary by P. Kirk

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