# Dimension bounds in monotonicity methods for the Helmholtz equation

## TL;DR

This paper improves bounds on the finite dimensional space in monotonicity methods for the Helmholtz equation, showing it is trivial when two coefficients have the same number of positive Neumann eigenvalues, enhancing shape detection techniques.

## Contribution

It refines the dimension bounds in monotonicity inequalities for the Helmholtz equation, providing conditions under which the finite dimensional space is trivial.

## Key findings

- Finite dimensional space bounds are improved.
- Trivial space when coefficients share the same positive Neumann eigenvalues.
- Enhances shape detection and inverse boundary problem methods.

## Abstract

The article [HPS] 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 $q_1 \leq q_2$, then the corresponding Neumann-to-Dirichlet operators satisfy $\Lambda(q_1) \leq \Lambda(q_2)$ up to a finite dimensional subspace. Here we improve the bounds for the dimension of this space. In particular, if $q_1$ and $q_2$ have the same number of positive Neumann eigenvalues, then the finite dimensional space is trivial.