Localization and Analytic Properties of the Solutions of the Simplest Quantum Filtering Equation

Abstract

The paper deals with the quantum Langevin equation describing a quantum particle with continuously observed position. Special Banach spaces of entire analytic functions are introduced and studied (including a theorem of Paley–Wiener type for them), which comprise all solutions of this equation and in which the uniform convergence (as time tends to infinity) of the solutions to the Gaussian function with a fixed dispersion (selflocalisation or continuous collapse) is proved. The asymptotic behavior at infinity of the mean position and momentum of the limit Gaussian wave packet (which satisfy classical Langevin equations) is also investigated.