Abstract <p> A special Cauchy problem for the Schrödinger equation with a delta potential localized on a half-line is addressed. From the viewpoint of high-frequency parabolic-equation heuristics, the problem could be viewed as an approximation to a paraxial (i.e., nearly tangential) diffraction of a plane wave incident on a screen. Explicit solution of the problem, found with the help of integral transformations, is subjected to exhaustive asymptotic investigation for all values of the complex coefficient of the potential. The asymptotic findings are qualitatively interpreted using diffraction terminology and quantitatively compared with the results of diffraction theory. Some effects that have no analogs in the related diffraction problems are noted. The solution is shown to be, in a certain range of parameters, an asymptotic solution of the Helmholtz equation with a delta potential. </p> <p> <b> DOI</b> 10.1134/S1061920825600679 </p>