  We consider the Dirichlet Laplace operator on open, quasi-bounded domains of
infinite volume. For such domains semiclassical spectral estimates based on the
phase-space volume - and therefore on the volume of the domain - must fail.
Here we present a method how one can nevertheless prove uniform bounds on
eigenvalues and eigenvalue means which are sharp in the semiclassical limit. We
give examples in horn-shaped regions and so-called spiny urchins. Some results
are extended to Schr\"odinger operators defined on quasi-bounded domains with
Dirichlet boundary conditions.
