Abstract

The article [B. Harrach, V. Pohjola, and M. Salo, Anal. PDE] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. The monotonicity inequality states that if two scattering coefficients satisfy q₁ ≤ q₂, then the corresponding Neumann-to-Dirichlet operators satisfy Λ(q₁) ≤ Λ(q₂) up to a finite-dimensional subspace. Here we improve the bounds for the dimension of this space. In particular, if q₁ and q₂ have the same number of positive Neumann eigenvalues, then the finite-dimensional space is trivial.