  We are concerned with the study of some classical spectral collocation
methods as well as with the new software system Chebfun in computing high order
eigenpairs of singular and regular Schrodinger eigenproblems. We want to
highlight both the qualities as well as the shortcomings of these methods and
evaluate them in conjunction with the usual ones. In order to resolve a
boundary singularity we use Chebfun with domain truncation. Although it is
applicable with spectral collocation, a special technique to introduce boundary
conditions as well as a coordinate transform, which maps an unbounded domain to
a finite one, are the special ingredients. A challenging set of \hard"benchmark
problems, for which usual numerical methods (f. d., f. e. m., shooting etc.)
fail, are analyzed. In order to separate \good"and \bad"eigenvalues we estimate
the drift of the set of eigenvalues of interest with respect to the order of
approximation and/or scaling of domain parameter. It automatically provides us
with a measure of the error within which the eigenvalues are computed and a
hint on numerical stability. We pay a particular attention to problems with
almost multiple eigenvalues as well as to problems with a mixed spectrum.
