  The vacuum (Casimir) energy in quantum field theory is a problem relevant
both to new nanotechnology devices and to dark energy in cosmology. The crucial
question is the dependence of the energy on the system geometry under study.
Despite much progress since the first prediction of the Casimir effect in 1948
and its subsequent experimental verification in simple geometries, even the
sign of the force in nontrivial situations is still a matter of controversy.
Mathematically, vacuum energy fits squarely into the spectral theory of
second-order self-adjoint elliptic linear differential operators. Specifically,
one promising approach is based on the small-t asymptotics of the cylinder
kernel e^(-t sqrt(H)), where H is the self-adjoint operator under study. In
contrast with the well-studied heat kernel e^(-tH), the cylinder kernel depends
in a non-local way on the geometry of the problem. We discuss some results by
the Louisiana-Oklahoma-Texas collaboration on vacuum energy in model systems,
including quantum graphs and two-dimensional cavities. The results may shed
light on general questions, including the relationship between vacuum energy
and periodic or closed classical orbits, and the contribution to vacuum energy
of boundaries, edges, and corners.
