# Lipschitz stability for a piecewise linear Schr\"{o}dinger potential from local Cauchy data

## TL;DR

This paper proves a global Lipschitz stability result for determining piecewise linear potentials in a Schrödinger equation from local boundary data, applicable to the reduced wave equation at fixed frequency, in dimensions three and higher.

## Contribution

It establishes the first Lipschitz stability estimate for piecewise linear potentials in the inverse Schrödinger problem without sign or spectral restrictions.

## Key findings

- Lipschitz stability holds for piecewise linear potentials in dimensions ≥ 3.
- The result applies to the reduced wave equation at fixed frequency.
- No sign or spectral conditions are required on the potential.

## Abstract

We consider the inverse boundary value problem of determining the potential $q$ in the equation $\Delta u + qu = 0$ in $\Omega\subset\mathbb{R}^n$, from local Cauchy data. A result of global Lipschitz stability is obtained in dimension $n\geq 3$ for potentials that are piecewise linear on a given partition of $\Omega$. No sign, nor spectrum condition on $q$ is assumed, hence our treatment encompasses the reduced wave equation $\Delta u + k^2c^{-2}u=0$ at fixed frequency $k$.