\abstract 
Given a self adjoint operator $A$ on a Hilbert space, 
suppose that that one wishes to compute the spectrum of 
$A$ numerically.  In practice, these problems
often arise in such a way that the matrix of $A$ relative
to a natural basis is ``sparse".  
For example, discretized second order differential operators
can be represented by doubly infinite tridiagonal matrices.  
In these cases it is easy and natural to compute the eigenvalues of large
$n\times n$ submatrices of the infinite operator matrix, and
to hope that if $n$ is large enough then the resulting distribution
of eigenvalues will give a good approximation to 
the spectrum of $A$.  Numerical analysts call this
the {\it Galerkin method}.  

While this hope is often realized in practice it
often fails as well, and it can fail in spectacular ways.  The
sequence of eigenvalue distributions may not converge as $n\to\infty$,
or they may converge to something that has little to do
with the original operator $A$.  At another level, even the 
meaning of `convergence' has not been made precise in general.  
In this paper we determine the proper general
setting in which one can expect convergence, and 
we describe the asymptotic behavior of the 
$n\times n$ eigenvalue distrubutions in all but the most
pathological cases.  Under appropriate hypotheses we establish
a precise limit theorem which shows how the spectrum of
$A$ is recovered from the sequence of eigenvalues of the
$n\times n$ compressions.  

In broader terms, our results have led us to the conclusion
that {\sl numerical problems involving infinite 
dimensional operators require a reformulation in terms of} 
\cstar s.  Indeed, it is only when the single 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 $n\times n$ eigenvalue distributions; the
limit is associated with a tracial state on $\Cal A$.  
Normally, $\Cal A$ is highly
noncommutative, and in our main applications it is a simple 
\cstar\ having a unique tracial state. 
We obtain precise asymptotic results for operators which 
represent discretized Hamiltonians of one-dimensional 
quantum systems with arbitrary continuous potentials.
\endabstract