On Schrِdinger operators perturbed by fractal potentials

Let Γ ⊂ Rn be a self-similar fractal. We discuss the problem of definition for the Schrödinger operators associated with the formal expression −Δβ,V,Γ = −Δ + βV, β ϵ R, where V is a generalized potential (distribution) supported by Γ and acting in the Sobolev scale, from W21(Rn) into W2−1(Rn). We give a precise sense to −Δβ,V,Γ as a self-adjoint operator in L2(Rn), present a qualitative characterization of its negative eigenvalues and prove that the limit −Δ∞,V,Γ = limβ→ ± ∞ − Δβ,V,Γ exists in the strong resolvent sense and coincides with the Friedrichs extension of the symmetric operator −Δ̇ = −Δ ∗|ƒ ϵ W22(Rn): ƒ ¦Γ = 0. In addition, we find conditions for −1 to be the lowest negative eigenvalue for −Δβ,V,Γ.