Improved stability of low Fourier modes in inverse problems for potentials
We prove Lipschitz, sub-H\"older and H\"older stability estimates for recovering the low Fourier modes of an unknown potential from the Dirichlet-to-Neumann (DN) map. We study three different cases, depending on the regularity of the difference $q_1-q_2$. First, we consider \[ (-\Delta - \lambda^2 + q)u=0 \quad \text{in} \quad \Omega\subset \mathbb{R}^n. \] We show that the difference $q_1-q_2$, assumed to be $M$-bandlimited, can be recovered in a Lipschitz stable way from the difference of the corresponding DN maps. This holds whenever $\lambda$ is sufficiently large relative to $M^{n/2}$. The proof involves real geometrical optics solutions. Secondly, we consider \[ (-\Delta + q)u=0 \quad \text{in} \quad (-\pi,\pi)^n. \] We show that the low Fourier coefficients of the difference $q_1-q_2$, assumed to be real-analytic and periodic, can be recovered in a sub-H\"older stable way from the difference of the corresponding DN maps. The number of recoverable Fourier modes grows as the DN maps become closer. Finally, we consider the case where the Fourier coefficients of the difference $q_1-q_2$ decay at a super-exponential rate $e^{-c|k|^{n/2}}$. We prove that the low Fourier modes can be recovered with H\"older stability, with the number of recoverable modes tending to infinity as the DN maps become closer. In all cases the $L^\infty$ potentials themselves do not need to satisfy additional assumptions or belong to a finite dimensional space. The constants in the stability estimates are uniform in the number of recovered Fourier modes.