Quantum motion on a halfline connected to a plane

Free motion of a particle on a manifold which consists of a one-dimensional and a two-dimensional parts connected in one point is treated. The class of admissible Hamiltonians is found using the theory of self-adjoint extensions. A particular attention is paid to those of them which allow the particle to pass through the point singularity; the reflection coefficient and other quantities characterizing scattering on the connection point are calculated. A possible physical application is discussed purpose the para-axial equations of motion, including charge exchange effects, has been studied for the system consisting of the quadrupole lenses. The dependence of final cross-over on beam current is discussed