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We show how a Bochner type formula can be used to establish universalinequalities for the eigenvalues of the drifted Cheng-Yau operator on a boundeddomain in a pinched Cartan-Hadamard manifold with the Dirichlet boundarycondition. In the first theorem, the hyperbolic space case is treated in anindependent way. For the more general setting, we first establish a Rauchcomparison theorem for the Cheng-Yau operator and two estimates associated withthe Bochner type formula for this operator. Next, we get some integralestimates of independent interest. As an application, we compute our universalinequalities. In particular, we obtain the corresponding inequalities for bothCheng-Yau operator and drifted Laplacian cases, and we recover the knowninequalities for the Laplacian case. We also obtain a rigidity result for aCheng-Yau operator on a class of bounded annular domains in a pinchedCartan-Hadamard manifold. In particular, we can use, e.g., the potentialfunction of the Gaussian shrinking soliton to obtain such a rigidity for theEuclidean space case. The fundamental gap conjecture is also addressed in thispaper.

Eigenvalue estimates of the drifted Cheng-Yau operator on bounded domains in pinched Cartan-Hadamard manifolds