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)?
Thanks in advance!
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.