The problem addressed in this paper is the reconstruction of Hamiltonians of quantum systems from observational data. Finding such ``causes'' or ``laws'' from a finite number of observations constitutes an inverse problem and is typically ill-posed in the sense of Hadamard [1-8].
Two research fields deal in particular with the reconstruction of potentials from spectral data (energy measurements): inverse spectral theory and inverse scattering theory. Inverse spectral theory characterizes the kind of data necessary, in addition to a given spectrum, to determine the potential [7, 9-13]. (See also Sect. 3.2.1.) Inverse scattering theory, in particular, considers, in addition to the spectrum, boundary data obtained `far away' from the scatterer. Those can be, for example, phase shifts obtained from scattering experiments [12,14,15].
In this paper, contrasting those two approaches, we will not exclusively be interested in spectral data, but will develop a formalism which allows to extract information from quite heterogeneous empirical data. In particular, we will consider in more detail the situation where the position of a quantum mechanical particle has been measured a finite number of times.
Due to increasing computational resources, the last decade has also seen a rapidly growing interest in applied empirical learning problems. They appear as density estimation, regression or classification problems and include, just to name a few, image reconstruction, speech recognition, time series prediction, and object recognition. Many disciplines, like applied statistics, artificial intelligence, computational and statistical learning theory, statistical physics, and also psychology and biology, contributed in developing a variety of learning algorithms, including for example smoothing splines [16], regularization and kernel approaches [4], support vector machines [17,18], generalized additive models [19], projection pursuit regression [20], expert systems and decision trees [21], neural networks [22], and graphical models [23].
Recently, their has been much work devoted to the comparison and unification of methods arising from different disciplines. (For an overview and comparison of methods see for example [24].) Hereby, especially the Bayesian approach to statistics proved to be useful as a unifying framework for empirical learning [22, 25-36].Bayesian approaches put special emphasis on a priori information which always has to accompany empirical data to allow successful learning.
The present paper is written from a Bayesian perspective. In particular, a priori information will be implemented in form of stochastic processes [37]. Compared to parametric techniques this has the advantage, that a priori information can typically be controlled more explicitly. Technically, this approach is intimately related to the well known Tikhonov regularization [2,3]. For an outline of the basic principles see also [38].
The paper is organized as follows: Sect. 2 gives a short introduction to Bayesian statistics. Sect. 3 applies the Bayesian approach to quantum mechanics and quantum statistics, with Sect. 3.1 concentrating on the treatment of empirical data for quantum systems and Sect. 3.2 discussing the implementation of a priori information. Sect. 3.3 presents two numerical case studies, the first dealing with the approximation of approximately periodic potentials, the second with inverse two-body problems. Sect. 4 shows how the approach can be applied to many-body systems, including the fundamentals of an inverse version of Hartree-Fock theory. Finally, Sect. 5 concludes the paper.