Very-high-precision solutions of a class of Schr{\"o}dinger equations

We investigate a method to solve a class of Schr{\"o}dinger equation eigenvalue problems numerically to very high precision $P$ (from thousands to a million of decimals). The memory requirement, and the number of high precision algebraic operations, of the method scale essentially linearly with $P$ when only eigenvalues are computed. However, since the algorithms for multiplying high precision numbers scale at a rate between $P^{1.6}$ and $P\,\log P\,\log\log P$, the time requirement of our method increases somewhat faster than $P^2$.