Upper and lower bounds on the mean square radius and criteria for the occurrence of quantum halo states

The root mean square (rms) radius is used in many fields of physics to characterize the size of quantum systems; this observable is thus of special interest. In the case of two-body systems, the rms radius and the energy of one eigenstate determine the depth and the range of the central potential which binds the particles provided that the number of bound states supported by the potential is also known. A simple example is the naive description of the deuteron by a squarewell potential (see, for example, Ref. [1]). The depth V0 and the range R can be adjusted to reproduce the rms radius and the binding energy of the deuteron. This adjustment is not unique except if the number of bound states in the potential is fixed to one. Obviously, these quantities are not sufficient to infer the shape of the potential. Indeed, the same simple description of the deuteron could be achieved with an exponential potential, for example. However, if information on the shape of the potential is obtained by other means, then constraints on the rms radius, on the energy, and on the number of bound states yield strong restrictions on the potential. Consequently, upper and lower limits on these quantities are interesting tools to obtain easily constraints on the interaction. There exists a fairly large number of upper and lower limits on the energy of eigenstates in the literature [2–10] as well as on the number of bound states supported by central potentials [3,11–17]. Similar results concerning the rms radius are scarcer. A first general inequality gives a lower limit on the rms radius of an ,-wave state in terms of the average kinetic energy s"=2m=1d ([18], p. 73)