Specific shape resonances were studied in systems with the highest point symmetry. The resonances occur in narrow energy intervals and disappear when inner potentials increase. A model system of point scatterers that are placed in the vertexes of an icosahedron and at its center was analyzed. In this system, the geometric resonance lifetime is extremely sensitive to structure changes and may attain 10−10 s, i.e., exceed the rotation period of electrons in atoms by six orders of magnitude. The feasibility of molecular traps that capture elastically scattered electrons and bring them into the resonant state is discussed. The resident time of electrons in these traps can be controlled with external static fields.