\abstract
This paper deals with mathematical issues relating to
the computation of spectra of self adjoint
operators on Hilbert spaces. We describe
a general method for approximating the spectrum
of an operator $A$ using the eigenvalues
of large finite dimensional
truncations of $A$. The results of several papers
are summarized which imply that the method is
effective in most cases of interest.
Special attention is paid to
the Schr\"odinger operators of one-dimensional
quantum systems.
We believe that these results serve to make
a broader point, namely that numerical problems involving
infinite dimensional operators
require a reformulation in terms of
\cstar s. Indeed, it is only when the given operator
$A$ is viewed as an element
of an appropriate \cstar\ $\Cal A$
that one can see the precise
nature of the limit of the finite dimensional
eigenvalue distributions: the
limit is associated with a tracial state on $\Cal A$.
For example, in the case where $A$ is the discretized
Schr\"odinger operator associated with a
one-dimensional quantum system, $\Cal A$ is a simple
\cstar\ having a unique tracial state. In these cases
there is a precise asymptotic result.
\endabstract