Advanced Mathematical Models & Applications

Goursat-type problem for the equation of small transverse vibrations of an infinite string with a concentrated force applied at the point is considered. The problem under consideration also covers the case when breaksoccur at points of the string. We study a solution to the problem under consideration that decreases for sufficiently large values of time. We prove the existence and uniqueness of the solution. Explicit formulas for the solution of the Goursat-type problem are obtained. In this paper, we demonstrate that the obtained results can be used to study transformation operators for the one-dimensional Schr¨odinger equation with delta-shaped potential. The Schr¨odinger equation with a delta-type potential arises when modeling situations with very narrow and sharp peaks or valleys of potential, for example, the interaction of a particle with a very thin barrier or attraction. Such a physical situation arises when there are very narrow and sharp maxima or minima of potential.