  We present a novel procedure to solve the Schr\"odinger equation, which in
optics is the paraxial wave equation, with an initial multisingular vortex
Gaussian beam. This initial condition has a number of singularities in a plane
transversal to propagation embedded in a Gaussian beam. We use the scattering
modes, which are solutions of the paraxial wave equation that can be combined
straightforwardly to express the initial condition and therefore permit to
solve the problem. To construct the scattering modes one needs to obtain a
particular set of polynomials, which play an analogous role than Laguerre
polynomials for Laguerre-Gaussian modes. We demonstrate here the recurrence
relations needed to determine these polynomials. To stress the utility and
strength of the method we solve first the problem of an initial Gaussian beam
with two positive singularities and a negative one embedded in. We show that
the solution permits one to obtain analytical expressions. These can used to
obtain closed expressions for meaningful quantities, like the distance at which
the positive and negative singularities merge, closing the loop of a vortex
line. Furthermore, we present an example of calculation of an specific
discrete-Gauss state, which is the solution of the diffraction of a
Laguerre-Gauss state showing definite angular momentum (that is, a highly
charged vortex) by a thin diffractive element showing certain discrete
symmetry. We show that thereby this problem is solved in a much simpler way
than using the previous procedure based in the integral Fresnel diffraction
method.
