We establish sharp semiclassical upper bounds for the moments of some negative powers for
the eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporated
in the problem, we obtain sharp lower bounds for the moments of positive powers not
exceeding one for such eigenvalues. When considering a Schrödinger operator with the
relativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove the
sharp Lieb-Thirring estimate for the moments of some negative powers of its
eigenvalues.