The need for a priori information

Typical results of inverse spectral theory show that, for example, a one-dimensional local potential can be reconstructed if a set of two complete spectra $\{E^{(1)}_\alpha\}_1^\infty$, $\{E^{(2)}_\alpha\}_1^\infty$ is given for two different boundary conditions for $\phi_\alpha$ [7,12]. Alternatively, a single spectrum is sufficient, if either a complete set of norming constants $u_\alpha$ = $\int \phi_\alpha dx$ is given (for a certain normalization of $\phi_\alpha$ which fixes the values of $\phi_\alpha$ on the boundary) [9] or the potential is already known on half of the interval [64]. Results from inverse scattering theory show under which circumstances a potential can be reconstructed from, e.g., a complete spectrum and the phase shifts as function of energy [12,14,15]. In practice, however, the number of actual measurements can only be finite. Thus, even if noiseless measurement devices would be available, an empirical determination of a complete spectrum, or of phase shifts as function of energy, is impossible. Therefore, to reconstruct a potential from experimental data in practice, inverse spectral or inverse scattering theory has to be combined with additional a priori information. If such a priori information is not made explicit -- as we try to do in the following -- it nevertheless enters any algorithm at least implicitly.

We address in this paper the measurement of arbitrary quantum mechanical observables, not restricted to spectral or scattering data. In particular, we have considered the measurement of particle positions. However, measuring particle positions only can usually not determine a quantum mechanical potential completely. For example, consider the ideal case of an infinite data limit $n\rightarrow\infty$ for a discrete $x$ variable (so derivatives with respect to $x$ have to be understood as differences) at zero temperature (i.e., $\beta\rightarrow\infty$). This, at least, would allow to obtain $p(x\vert\hat x,v)$ = $\vert\phi_0(x)\vert^2$ to any desired precision. But even when we restrict to the case of a local potential, we would also need, for example, the ground state energy $E_0$ and $\phi_0^*(x)\phi^{\prime\prime}_0(x)$ to determine $v(x)$ from the eigenvalue equation of $H$

$$v(x) = E_0+\frac{1}{2m} \frac{\phi_0^*(x)\phi^{\prime\prime}_0(x)}{\vert\phi_0(x)\vert^2} ,$$ (32)

where $\phi_0^{\prime\prime}$ = $\partial^2\phi_\alpha(x)/\partial x^2$ (or a discretized version thereof). For finite data, a nonlocal potential, continuous $x$, or finite temperature the situation is obviously even worse. In the high temperature limit, for example, $p(x\vert\hat x,v)$ becomes uniform and independent from the potential. Summarizing, even in the ideal case where the complete true likelihood $p(x\vert\hat x,v)$ is assumed to be known, the problem of reconstructing potential can still be ill-posed. (The corresponding time-dependent problem, i.e., the reconstruction of a potential $v$ given the complete time-dependent likelihood, is treated in [65]. A Bayesian approach for time-dependent systems, based on finite data, can be found in [66].) Hence, while a priori information is crucial for every learning problem [32,33,67], the reconstruction of a quantum mechanical potential is particularly sensitive to the implemented a priori information.