Get Analytic Deformations of the Spectrum of a Family of Dirac PDF

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 lower than an analytic perturbation of the operator. different types 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 enormous collar connected to its boundary.

The unifying concept at the back of the research of those varieties of spectra is the idea of definite "eigenvalue-Lagrangians" within the symplectic house $L^2(\partial M)$, an idea 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 different types of spectra which go through 0 behave in primarily a similar approach (to first non-vanishing order). occasionally, this ends up in topological algorithms for computing spectral move.

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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

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We introduce the notation Mx(t) = G(t)U(X,t)NQ(0) 54 P. KIRK AND E. KLASSEN and Cx(t) = GitM^ty'UiX^NoiO); these are analytically varying Lagrangians commensurate to NQ(Q). C0(t) = M0(t) for any £, and that C 0 (0) = Afo(0) = N0(0). Consider the symplectic reduction Note that p:C^£(V®JV) defined by p{K) Y^w = • Since N0(0) f l ^ = 0, M\(t) and C\(i) are transverse to W for A, t close to 0, p(M\(t)) and /P(CA(£)) vary analytically in the symplectic space V 0 JV. Since p(M 0 (0)) = p(C 0 (0)) = p(N0(0)) = V, p(MA(*)) and p(CA(*)) are "graphs" of self-adjoint linear maps on V.

The point ((£, A), M) eU x £ lies in B if the intersection of M with •P\~(t) is non-zero, and it lies in V(L(t)) if the intersection with L(t) 0 -P\"(t) is non-zero. Notice that B C V(L(t)) for any path L(t), and that the discontinuities of K lie along S = N~l(B). Indeed N~1(B) is exactly the union of the graphs of the type 1 eigenvalues, and N~l(V(L(t)) is the union of the graphs of all the eigenvalues. 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)).

3 Relation to weighted L2 eigenvalues The small extended L2 eigenvalues which respect L can be interpreted in terms of the real eigenvalues of Fredholm operators acting on weighted Sobolev spaces in the sense of Lockhart and McOwen [LM]. With D as before, let 6 > 0 be smaller than half of the smallest positive eigenvalue of the tangential operator A. Extend D in the obvious way to X(oo). Choose a smooth function r : X(oo) —* [0, oo) such that r is the projection onto the second factor on Y x [l,oo), and r = 0 on X(0).

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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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