Phase space deformation and basis set optimization

By deforming a given region of phase space---occupied by some unknown eigenfunctions one wishes to find---into a standard, integrable region, effective reductions in basis set size can be achieved. In one-dimensional problems we are able to achieve B/C=1+O(Planck' s constant), where B is the basis set size and C is the number of "converged" eigenfunctions. This result is confirmed by numerical examples, which also indicate exponential convergence as the basis set size is increased. In higher dimensions we prove that such an optimistic result is impossible; we expect that the best one can do in this case is B/C=a+o(1), where a>1 has a geometric interpretation in terms of ratios of volumes in phase space.