  We set-up and solve the Cauchy problem for Schr\"odinger-type differential
operators with generalized functions as coefficients, in particular, allowing
for distributional coefficients in the principal part. Equations involving such
kind of operators appeared in models of deep earth seismology. We prove
existence and uniqueness of Colombeau generalized solutions and analyze the
relations with classical and distributional solutions. Furthermore, we provide
a construction of generalized initial values that may serve as square roots of
arbitrary probability measures.
