We consider the self-adjoint Schrödinger operator Aα in L2(R^d), d ≥ 2, with
a δ-potential supported on a Lipschitz hypersurface Σ ⊆ Rd of strength α ∈Lp(Σ)+L∞(Σ). Formally, this operator is given by
We show the uniqueness of the ground state and, under some additional conditions on the coefficient α and the hypersurface Σ, we determine the essential spectrum of Aα. In the special case that Σ is a hyperplane, we obtain a
Birman-Schwinger principle with a relativistic Schrödinger operator
in L2(R^(d−1)), as Birman-Schwinger operator. As an application we prove an optimization result for the bottom for the spectrum of Aα .