Essential spectrum of schrödinger operators with δ and δ^′-interactions on systems of unbounded smooth hypersurfaces in Rⁿ

The purpose of this paper is to study of the essential spectra of Schrödinger operators on ℝⁿ with surface δ and δ^′-type interactions on systems (formula presented) of simply connected unbounded C²-hypersurfaces in ℝⁿ (dim Γ_k = n−1) such that dist(Γ_k, Γ_l ) > 0 for k ≠ l. We consider the formal Schrödinger operators (formula presented) where δ_Γj are the delta-functions supported on the hypersurfaces Γ_j, δΓ^′_j are the normal derivatives of δ_Γj . We show in the paper that the addition to the potential W ∈ L^∞(ℝⁿ ) of delta-type potentials supported on unbounded hypersurfaces significantly changes the essential spectrum of the Schrödinger operators −Δ + W, and we give an effective description of the essential spectra of Schrödinger operators (1) taking into account a behavior at infinity the hypersurfaces Γ_j, the potentials W, and the strength interaction coefficients α_j, β_j, j = 1, …, N. We apply for the study of the essential spectra of operators H^{, H}_δΓ,α δ Γ^′,β the limit operators method adapted for investigations of unbounded operators.