Maslov's canonical operator, Hörmander's formula, and localization of the Berry–Balazs solution in the theory of wave beams

For three-dimensional Schrödinger equations, we study how to localize exact solutions represented as the product of an Airy function (Berry–Balazs solutions) and a Bessel function and known as Airy–Bessel beams in the paraxial approximation in optics. For this, we represent such solutions in the form of Maslov's canonical operator acting on compactly supported functions on special Lagrangian manifolds. We then use a result due to Hörmander, which permits using the formula for the commutation of a pseudodifferential operator with Maslov's canonical operator to “move” the compactly supported amplitudes outside the canonical operator and thus obtain effective formulas preserving the structure based on the Airy and Bessel functions. We discuss the influence of dispersion effects on the obtained solutions.