On Schrodinger operators perturbed by fractal potentials

Let Gamma subset of R-n be a self-similar fractal. We discuss the problem of definition for the Schrodinger operators associated with the formal expression -Delta(beta,V,Gamma) = -Delta + beta V, beta epsilon R, where V is a generalzed potential (distribution) supported by Gamma and acting in the Sobolev scale, from W-2(1)(R-n) into W-2(-l) (R-n). We give a precise sense to -Delta(beta,V,Gamma) as a self-adjoint operator in L-2 (R-n), present a qualitative characterization of its negative eigenvalues and prove that the limit -Delta(infinity,V,Gamma) = lim(beta-->+/-infinity) -Delta(beta,V,Gamma) exists in the strong resolvent sense and coincides with the Friedrichs extension of the symmetric operator -(Delta) over dot = -Delta \ {f epsilon W-2(2) (R-n) : f \ Gamma = 0}. In addition, we find conditions for -1 to be the lowest negative eigenvalue for -Delta(beta,V,Gamma).