# spectral sections - explicit construction

Melrose and Piazza defined the concept of "spectral sections" (see Journal of Differential Geometry, 45 (1997), p.99-180).

I am now looking for nontrivial examples and methods to explicitely construct such sections. Does anybody know references for that (if they exist)?

PS: Maybe a short description of this concept would be helpful:

Take a family of self-adjoint differential operators $D_\beta$ of first order, parametrized over a compact base space B. Then every $D_\beta$ has a discrete spectrum with finite dimensional eigenspaces. Let $\Pi_\beta$ be the projection onto the eigenspaces with positive eigenvalues. $\Pi_\beta$ is in general not continuous in the variable $\beta$. A spectral section now is a family $P_\beta$ of projections, continuously depending on $\beta$, so that $P_\beta - \Pi_\beta$ is a compact operator for every $\beta$.

The existence of spectral sections can be determined using some kind of index in K-theory. This is nice, but does not tell you much about the construction of spectral sections.

You probably already know this, but the standard method is to assume there is a spectral gap, i.e. a real number not in the spectrum of $D_\beta$ for all $\beta$. So although, as you point out, zero might not work, another real number might, or more generally, if there is a function $g:B\to R$ so that $g(\beta)$ does not lie in the spectrum of $D_\beta$ then you can use it to construct a spectral section.
Another general method, when the space $B$ is an interval and the dependence of $D_\beta$ on $\beta$ is analytic, is to use the Kato selection lemma, which allows you to analytically continue eigenvectors, and in particular if you take the positive eigenspan at one parameter, you can extend this to a spectral section over the interval, even if there is no spectral gap. Presumably if you are careful you can extend this idea for more general analytic families. Typically families which arise in geometric contexts vary analytically, although this is not always easy to check.
• A couple of thoughts: if the spectral projections $\Pi_\beta$ are all compact perturbations of each other, you are considering maps from B to the Grassmanian of projections that differ from a fixed $\Pi_{\beta{0}}$ by a compact operator. There is a literature on this: e.g. Chapter 15 of Booss-Wociechowski. Another thought: if your family of Dirac operators is a family on $\partial M$ and you can extend them over $M$ then the Calderon projectors for the extensions (explained in [BW]) gives you a spectral section, and varying the operators on the interior gives you many. – Paul Feb 17 '10 at 16:13