Bayesian reconstruction of approximately periodic potentials for quantum systems at finite temperature

The paper discusses the reconstruction of potentials for quantum systems at finite temperatures from observational data. A nonparametric approach is d eveloped, based on the framework of Bayesian statistics, to solve such inve rse problems. Besides the specific model of quantum statistics giving the p robability of observational data, a Bayesian approach is essentially based on a priori information available for the potential. Different possibilitie s to implement a priori information are discussed in detail, including hype rparameters, hyperfields, and non-Gaussian auxiliary fields. Special emphas is is put on the reconstruction of potentials with approximate periodicity. Such potentials might for example correspond to periodic surfaces modified by point defects and observed by atomic force microscopy. The feasibility of the approach is demonstrated for a numerical model.