Born's postulate and reconstruction of the \(\psi\)-function in nonrelativistic quantum mechanics
\textit{W. Pauli} [Handbuch der Physik, Springer-Verlag, 1958, Bd. 5, footnote on p. 17] asks to what extent the absolute values \(| \psi (x)|\) and \(| {\hat \psi}(p)|\) determine the function \(\psi\) :\({\mathbb{R}}\) \(3\to {\mathbb{C}}\), where \({\hat \psi}\) denotes the Fourier- transform of \(\psi\). Generalising this set-up one may ask whether the data \[ (*)\quad \{\| P_ A(E)x\| | A\in {\mathcal A},E\in {\mathcal B}({\mathbb{R}})\} \] for a fixed set \({\mathcal A}\) of self-adjoint operators in a complex Hilbert space X determine x in X up to a constant factor (here \(P_ A(\cdot)\) is the projection-valued measure corresponding to A, and \({\mathcal B}({\mathbb{R}})\) is the set of Borel subsets of \({\mathbb{R}})\). Developing an idea going back to V. E. Katsnel'son [cf., the reviewer, Zap. Nauchn. Sem. Leningrad. Otd. Math. Inst. Steklov 39, 189-190 (1974; Zbl 0353.47014)] the author constructs an infinite system of operators \({\mathcal A}=\{A(\alpha)| \alpha \in {\mathbb{R}}\}\) such that A(\(\alpha)\) and A(\(\beta)\) have no common non-trivial invariant subspace when \(\alpha\neq \beta\) and the data (*) do not determine x up to a constant factor, thereby disproving a reviewer's conjecture. Moreover, improving on the previous work of \textit{V. Ya. Kreinovic} [see Theoretical and Mathematical Physics 28, 628-633 (1977)] the author shows that position measurements carried out at different moments of time on an ensemble of particles placed in the potential \[ H(\rho)=-\Delta -\sum^{n}_{j=1}\rho_ jx^ 2_ j \] provide sufficient data to reproduce the initial state of the ensemble if \(\sqrt{\rho_ j}-\sqrt{\rho_ i}\) is not a rational number when \(i\neq j\).
scientific article; zbMATH DE number 4032400