Dimension bounds in monotonicity methods for the Helmholtz equation

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 \leq q2, then the corresponding Neumann-to-Dirichlet operators satisfy \Lambda (q1) \leq \Lambda (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.