Robustness of a quasiparticle interference test for sign-changing gaps in multiband superconductors

Recently, a test for a sign-changing gap function in a candidate multiband unconventional superconductor involving quasiparticle interference data was proposed. The test was based on the antisymmetric, Fourier transformed conductance integrated over a range of momenta $\mathbf{q}$ corresponding to interband processes, which was argued to display a particular resonant form, provided the gaps changed sign between the Fermi-surface sheets connected by $\mathbf{q}$. The calculation was performed for a single impurity, raising the question of how robust this measure is as a test of sign-changing pairing in a realistic system with many impurities. Here we reproduce the results of the previous work within a model with two distinct Fermi-surface sheets, and we show explicitly that the previous result, while exact for a single nonmagnetic scatterer and also in the limit of a dense set of random impurities, can be difficult to implement for a few dilute impurities. In this case, however, appropriate isolation of a single impurity is sufficient to recover the expected result, allowing a robust statement about the gap signs to be made.