  We study the approximation of the smallest eigenvalue of a Sturm-Liouville
problem in the classical and quantum settings. We consider a univariate
Sturm-Liouville eigenvalue problem with a nonnegative function $q$ from the
class $C^2([0,1])$ and study the minimal number $n(\e)$ of function evaluations
or queries that are necessary to compute an $\e$-approximation of the smallest
eigenvalue. We prove that $n(\e)=\Theta(\e^{-1/2})$ in the (deterministic)
worst case setting, and $n(\e)=\Theta(\e^{-2/5})$ in the randomized setting.
The quantum setting offers a polynomial speedup with {\it bit} queries and an
exponential speedup with {\it power} queries. Bit queries are similar to the
oracle calls used in Grover's algorithm appropriately extended to real valued
functions. Power queries are used for a number of problems including phase
estimation. They are obtained by considering the propagator of the discretized
system at a number of different time moments. They allow us to use powers of
the unitary matrix $\exp(\tfrac12 {\rm i}M)$, where $M$ is an $n\times n$
matrix obtained from the standard discretization of the Sturm-Liouville
differential operator. The quantum implementation of power queries by a number
of elementary quantum gates that is polylog in $n$ is an open issue.
