Yes, the stability and convergence analysis can be significantly improved by incorporating additional a priori information about the potential or the measurement data. By utilizing prior knowledge, such as bounds on the potential or smoothness assumptions, one can refine the regularization parameter selection, leading to enhanced stability in the reconstruction process.
Incorporating a priori information can also help in formulating more robust Tikhonov regularization functionals that account for the specific characteristics of the potential, such as sparsity or boundedness. This can lead to better convergence rates and reduced sensitivity to noise in the measurement data. Furthermore, advanced techniques such as Bayesian inference can be employed to integrate prior distributions of the potential, allowing for a probabilistic interpretation of the reconstruction process. This approach not only improves the stability and convergence of the numerical method but also provides a framework for quantifying uncertainty in the reconstructed potential.