Schroedinger's equation is just written as an eigenvalue problem. H could be anything.

And exp(i pi) = -1 is a definition, not an equation (equivalent to i^2 = -1). Also there are infinitely many such definitions, for the various Clifford and Grassmann algebras.

I can't believe neither GR:

Rmn = -(8 pi G) Tmn

nor the Dirac equation:

[ ym (dm + ieAm) + iM ] psi = 0

were mentioned!