Can Milne's method work well for the Coulomb-like potentials ?

摘要

We extend the improved Milne's (Milne-spline) method for obtaining eigenvalues and eigenfunctions to the cases of long-range and singular potentials, for which we have conjectured that it is difficult to apply the method. Contrary to our conjecture it turned out that the method is valid also for Coulomb potential and repulsive 1/xn(n=2,3,…) type potential. Further we applied the method for two cases, for which the solutions are not known, in order to investigate the stability of the multi-dimensional universe. It has been shown that the extra-dimensional (internal) space of our universe is not stable in classical Einstein gravity as well as canonically quantized one. Two possibilities for stabilization were investigated: (i) noncanonically quantized Einstein gravity and (ii) canonically quantized higher curvature gravity. It has been suggested that the space is stable by qualitative and approximate methods. Exact analytical treatment is very difficult, so that numerical investigation is highly desirable. Numerical investigation shows that the space is stable with sufficient reliability.