  We show how a Bochner type formula can be used to establish universal
inequalities for the eigenvalues of the drifted Cheng-Yau operator on a bounded
domain in a pinched Cartan-Hadamard manifold with the Dirichlet boundary
condition. In the first theorem, the hyperbolic space case is treated in an
independent way. For the more general setting, we first establish a Rauch
comparison theorem for the Cheng-Yau operator and two estimates associated with
the Bochner type formula for this operator. Next, we get some integral
estimates of independent interest. As an application, we compute our universal
inequalities. In particular, we obtain the corresponding inequalities for both
Cheng-Yau operator and drifted Laplacian cases, and we recover the known
inequalities for the Laplacian case. We also obtain a rigidity result for a
Cheng-Yau operator on a class of bounded annular domains in a pinched
Cartan-Hadamard manifold. In particular, we can use, e.g., the potential
function of the Gaussian shrinking soliton to obtain such a rigidity for the
Euclidean space case. The fundamental gap conjecture is also addressed in this
paper.
