On orthogonalizing of the system of functions with minimal Heisenberg uncertainty

Abstract:
The total normed (but not orthonormal) system of functions (coherent states) with minimal Heisenberg uncertainty is considered. In 1932 J. fon Neuman had set up a hypothesis that orthogonalizing this system in arbitrary way we get a total orthonormal system of functions with uniformly bounded Heisenberg uncertainty. However, up to now this hypothesis remain open. In this paper the proof of some (partial) analogue of the J. fon Neumanstate is suggested.