  We consider Laplacian Growth of self-similar domains in different geometries.
Self-similarity determines the analytic structure of the Schwarz function of
the moving boundary. The knowledge of this analytic structure allows us to
derive the integral equation for the conformal map. It is shown that solutions
to the integral equation obey also a second order differential equation which
is the one dimensional Schroedinger equation with the sinh inverse square
potential. The solutions, which are expressed through the Gauss hypergeometric
function, characterize the geometry of self-similar patterns in a wedge. We
also find the potential for the Coulomb gas representation of the self-similar
Laplacian growth in a wedge and calculate the corresponding free energy.
