Singularly continuous spectrum of singularly perturbed operators

We propose a construction of a singularly perturbed self-adjoint operator with a given compact set in its singularly continuous spectrum. In particular, the set can be a fractal of prescribed type. We use the construction of a singularly perturbed operator à for a given self-adjoint operator A in a Hilbert space $\mathcal{H}$ that solves the eigenvalue problem Ãψ _i = λ _iψ_i for a countable set Λ = {λ _i} _i=1 ^∞ of real numbers λ _i ∈ ?¹, |λ _i| < ∞, and an orthonormal system of vectors {ψ _i}, i = 1, 2 …, under certain additional general conditions.