ua nt - p h / 05 08 21 3 v 1 2 9 A ug 2 00 5 Localization for free ultrametric quantum particle

Free evolution for quantum particle in generic ultrametric space is considered. We prove that if mean zero wave packet is localized in some space domain then its evolution remains localized in the same domain. In the present note we consider ultrametric quantum mechanics with real time and ultrametric space. Quantum mechanics with ultrametric (p–adic and adelic) space was considered in many works, see for example [1]–[9]. We consider the ultrametric quantum particle propagating generic ultrametric space, which in general has no any group structure, with the dynamics described by the ultrametric Schrodinger equation (1) below. The main observation of the present paper (which is valid also for the particular p–adic case) is that ultrametricity of the space implies the new phenomenon — localized in space mean zero wave packets remains localized for any moment of time. Here we assume that time is real but the same observation is valid for the case when both space and time are ultrametric. The model under consideration is as follows (see the Appendix for the details). Consider product X ×R of ultrametric space X of the analytic type and the real line R, with the coordinates x and t correspondingly. Consider ultrametric pseudodifferential operator D x of the sup–type (see the Appendix), acting on functions in L 2 (X, ν), and the ultrametric Schrodinger equation with real time i¯ h∂ t − ¯ h 2 D x Ψ(x, t) = 0 (1) Equation (1) describes free quantum particle in ultrametric space and real time. One can also consider ultrametric quantum particle in potential, described by the following ultrametric Schrodinger equation i¯ h∂ t − ¯ h 2 D x − U(x) Ψ(x, t) = 0