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 , is given for two different boundary conditions for [7,12]. Alternatively, a single spectrum is sufficient, if either a complete set of norming constants = is given (for a certain normalization of which fixes the values of 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
for a discrete variable
(so derivatives with respect to have to be understood as differences)
at zero temperature (i.e.,
).
This, at least, would allow to obtain
=
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
and
to determine from the eigenvalue equation of
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