  The goal of this paper is to derive estimates of eigenvalue moments for
Dirichlet Laplacians and Schr\"odinger operators in regions having infinite
cusps which are geometrically nontrivial being either curved or twisted; we are
going to show how those geometric properties enter the eigenvalue bounds. The
obtained inequalities reflect the essentially one-dimensional character of the
cusps and we give an example showing that in an intermediate energy region they
can be much stronger than the usual semiclassical bounds.
