# Download Analytic Deformations of the Spectrum of a Family of Dirac by P. Kirk PDF

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, relatively, how this spectrum varies less than an analytic perturbation of the operator. sorts of eigenfunctions are thought of: first, these pleasurable 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 thought at the back of the research of those varieties of spectra is the suggestion of sure "eigenvalue-Lagrangians" within the symplectic house $L^2(\partial M)$, an idea as a result of Mrowka and Nicolaescu. via learning the dynamics of those Lagrangians, the authors may be able to identify that these parts of the 2 varieties of spectra which go through 0 behave in primarily an identical means (to first non-vanishing order). every so often, this ends up in topological algorithms for computing spectral move.

Read or Download Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold With Boundary PDF

Similar science & mathematics books

Prescribing the curvature of a Riemannian manifold

Those notes have been the foundation for a sequence of ten lectures given in January 1984 at Polytechnic Institute of latest York lower than the sponsorship of the convention Board of the Mathematical Sciences and the nationwide technological know-how beginning. The lectures have been aimed toward mathematicians who knew both a few differential geometry or partial differential equations, even if others may well comprehend the lectures.

Design and Nature V: Comparing Design in Nature with Science and Engineering

Nature has proven a rare ability to boost dynamic constructions and structures over many thousands of years. What researchers examine from those buildings and platforms can usually be utilized to enhance or advance human-made constructions and platforms. and there's nonetheless a lot to be discovered. geared toward supplying clean impetus and thought for researchers during this box, this ebook comprises papers awarded on the 5th foreign convention on layout and Nature.

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

Example text

Let (^) f c , / ^ , etc. always denote the fcth derivative at 0. If i>o, ^o £ V^, choose t;*, v^ i = 1, • • • , n — 1 as in the definition of Vn. Then 0=< £(*)(£ ^ ) , ^ f i > - < ^Vit^DWfcvit*) t >. i i Taking nth derivatives and using the definition of Vn, we get < (^)B(Z>(t)(5>t<)),«o >=< vo,(£)nW)(Eeiti)) > • i i Thus Bn(vo,vo) is independent of the choice of Vi for i > 0, and hence J5n is well-defined. Since the inner product is Hermitian, so is Bn. It is trivial to check that V^+i C ker£?

3. Decompose Sk into the direct sum Sk = S£x 0 S^ A by taking s £\ = s P an ivk,\} and 5 s fc,A = 4. Let Pan iv-k,\}- W = kerA, -P\ = P\ = ®k>nSkX ®k>nSkfX (by which we mean the closure in L2(E) of this sum), so that L2(£) = PA-0W0PA+. 1) Notice that we have avoided the case A = /i* 7^ 0. We will chiefly be interested in small eigenvalues; in particular it suits our purpose to assume that |A| is smaller than the smallest positive eigenvalue /xn+i of the tangential operator A. In practice, we will consider small A.

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. The next lemma shows that P^(t) vary analytically in t and A. 1 by working on L2(Y x (—oo, 0]). 1]. (We wish to thank U. 3 LEMMA. The subspaces PJ^it) vary analytically in t and A. Moreover, there exists an analytically varying family of (non-orthogonal) projections H\(t) : L2(E) -> P+(t) so that nx(t){avk,x(t) + bv-klX(t)) = avktX(t) for k> 0. Proof. 1 of [KK4] for details.