Optimization of the lowest eigenvalue for the Schr\"odinger operator with a $\delta$-potential supported on a hyperplane

We consider the self-adjoint Schrödinger operator in L(R), d ≥ 2, with a δ-potential supported on a hyperplane Σ ⊆ R of strength α = α0+α1, where α0 ∈ R is a constant and α1 ∈ L(Σ) is a nonnegative function. As the main result, we prove that the lowest spectral point of this operator is not smaller than that of the same operator with potential strength α0+α∗1 , where α ∗ 1 is the symmetric decreasing rearrangement of α1. The proof relies on the Birman-Schwinger principle and the reduction to an analogue of the Pólya-Szegő inequality for the relativistic kinetic energy in R.