  Given a Schr\"odinger operator with a real-valued potential on a bounded,
convex domain or a bounded interval we prove inequalities between the
eigenvalues corresponding to Neumann and Dirichlet boundary conditions,
respectively. The obtained inequalities depend partially on monotonicity and
convexity properties of the potential. The results are counterparts of
classical inequalities for the Laplacian but display some distinction between
the one-dimensional case and higher dimensions.
