Soliton-like solutions to the ordinary Schroedinger equation within standard quantum mechanics

In recent times attention has been paid to the fact that (linear) wave equations admit of 'soliton-like' solutions, known as localized waves or non-diffracting waves, which propagate without distortion in one direction. Such localized solutions (existing also for K-G or Dirac equations) are a priori suitable, more than gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, localized solutions exist even for the ordinary (linear) Schroedinger equation within standard quantum mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions (even if localized and 'decaying') are not square-integrable, as well as plane or spherical waves: we show therefore how to obtain finite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential.