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This paper presents a posteriori error estimates for conforming numericalapproximations of eigenvalue clusters of second-order self-adjoint ellipticlinear operators with compact resolvent. Given a cluster of eigenvalues, weestimate the error in the sum of the eigenvalues, as well as the error in theeigenvectors represented through the density matrix, i.e., the orthogonalprojector on the associated eigenspace. This allows us to deal with degenerate(multiple) eigenvalues within the framework. All the bounds are valid under theonly assumption that the cluster is separated from the surrounding smaller andlarger eigenvalues; we show how this assumption can be numerically checked. Ourbounds are guaranteed and converge with the same speed as the exact errors.They can be turned into fully computable bounds as soon as an estimate on thedual norm of the residual is available, which is presented in two particularcases: the Laplace eigenvalue problem discretized with conforming finiteelements, and a Schr{\"o}dinger operator with periodic boundary conditions ofthe form $--$\Delta$ + V$ discretized with planewaves. For these two cases,numerical illustrations are provided on a set of test problems.

Guaranteed a posteriori bounds for eigenvalues and eigenvectors: multiplicities and clusters