The Negative Spectrum of Schr?dinger Operators with Fractal Potentials

Let Γ?R~2 be a regular anisotropic fractal. We discuss the problem of the negative spectrum for the Schr?dinger operators associated with the formal expression H_β=id-?+βtr_b~Γ,β∈R,acting in the anisotropic Sobolev space W_2~(1,α)(R~2), where ? is the Dirichlet Laplanian in R~2 and tr_b~Γ is a fractal potential(distribution) supported by Γ.