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!

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.