For a compact Riemann surface split by quasicircles into connected pieces of genuses g1 and g2, the index of the Schiffer operator T_{1,2} equals g1 minus g2.
On holomorphick-differentials on open Riemann surfaces
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Scattering theory on Riemann surfaces I: Schiffer operators, cohomology, and index theorems
For a compact Riemann surface split by quasicircles into connected pieces of genuses g1 and g2, the index of the Schiffer operator T_{1,2} equals g1 minus g2.