If we have a sequence of linear maps and finite dimensional inner product spaces $$X\xrightarrow{f} Y\xrightarrow{g}Z$$ such that $g\circ f=0$, then we can consider the Hodge-Laplace operator $$\Delta:=f\circ f^t+g^t\circ g:Y\to Y.$$ A classical observation is that $$\ker g=\operatorname{img}f\oplus\ker\Delta,$$ so that the map $\ker\Delta\to H:=\ker g/\operatorname{img}g$ induced by the inclusion $\ker\Delta\hookrightarrow\ker g$ is an isomoprhism. This is how this construction — or rather its infinitie dimensional, t.v.s. version — becomes useful, for example, in Hodge theory.

Now the operator $\Delta$ depends on the inner products on the three vector spaces $X$, $Y$ and $Z$, even though the dimension of its kernel does not.

Question. Is it known how the spectra of the operator varies when one changes those inner products?

I am particularly interested in the finite dimensional situation. In the original context of differential forms and so on one studies the asymtotic growth of eigenvalues and so on, and it turns out one can control that, but in the finite dimensional situation there is not much growth to speak of.