# Pseudo-differential analysis of the Helmholtz layer potentials on open curves

## TL;DR

This paper develops a pseudo-differential operator framework on open curves to analyze Helmholtz layer potentials, enabling the construction of efficient preconditioners for 2D scattering problems involving screens.

## Contribution

It introduces new classes of pseudo-differential operators on open curves and applies symbolic calculus to analyze Helmholtz layer potentials, facilitating preconditioner design.

## Key findings

- Established symbolic calculus for new operator classes
- Constructed low order parametrices as square roots of tangential operators
- Provided theoretical foundation for efficient Helmholtz preconditioners

## Abstract

We introduce two new classes of pseudo-differential operators on open curves. They correspond via a change of variables to subclasses of the periodic pseudo-differential operators, which respectively stabilize even and odd functions. The resulting symbolic calculus can be applied to the analysis of the Helmholtz weighted layer potentials on open curves. In particular, we build some low order parametrices of the layer potentials which take the form of square roots of tangential operators. This gives some foundation for the construction of efficient preconditioners for the Helmholtz scattering problem by a screen in 2D.