Gheorghiu, C.-I. Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations. Computation2021, 9, 2.
Gheorghiu, C.-I. Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations. Computation 2021, 9, 2.
Gheorghiu, C.-I. Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations. Computation2021, 9, 2.
Gheorghiu, C.-I. Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations. Computation 2021, 9, 2.
Abstract
We are concerned with the use of some classical spectral collocation
methods as well as with the new software system Chebfun in order to compute
high order (index) eigenpairs of singular as well as regular Schrodinger
eigenproblems.
We want to highlight both the qualities as well as the shortcomings of these
methods and evaluate them vis-a-vis the usual ones.
In order to resolve a boundary singularity we use Chebfun with the simple
domain truncation technique. Although this method is equally easy to apply
with spectral collocation, things are more nuanced in the case of these methods.
A special technique to introduce boundary conditions as well as a coordinate
transform which maps an unbounded domain to a nite one are the ingredients.
A challenging set of "hard" benchmark problems, for which usual numerical
methods (f. d., f. e. m., shooting etc.) fail, are analysed. 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 for problems with a mixed (continuous) spectrum. In the latter case
we try to numerically highlight its existence.
Special attention will be paid to the higher eigenpairs (the pair of eigenvalue
and the corresponding eigenfunction approximated by an eigenvector spanning
its nodal values).
Keywords
spectral collocation; Chebfun; singular Schrodinger; high index eigenpairs; multiple eigenpairs; accuracy; numerical stability
Subject
Computer Science and Mathematics, Mathematics
Copyright:
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