On the Computation of the Eigenproblems of Hydrogen and Helium in Strong Magnetic and Electric Fields with the Sparse Grid Combination Technique

We introduce the combination technique for the numerical solution of d-dimensional eigenproblems on sparse grids. Here, O(d·(log N)d-1) different problems, each of size O(N), have to be solved independently. This is in contrast to the one problem of size O(Nd) for a conventional finite element discretization, where N denotes the number of grid points in one coordinate direction. Therefore, also higher dimensional eigenvalue problems can be treated by our sparse grid combination approach. We apply this method to solve the three-dimensional Schrِdinger equation for hydrogen (one-electron problem) and the six-dimensional Schrِdinger equation for helium (two-electron problem) in strong magnetic and electric fields.