  Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the
$m$-th Laplace operator with a recently introduced extra-stabilized
nonconforming Crouzeix-Raviart ($m=1$) or Morley ($m=2$) finite element
eigensolver. Striking numerical evidence for the superiority of a new adaptive
eigensolver motivates the convergence analysis in this paper with a proof of
optimal convergence rates of the GLB towards a simple eigenvalue. The proof is
based on (a generalization of) known abstract arguments entitled as the axioms
of adaptivity. Beyond the known a priori convergence rates, a medius analysis
is enfolded in this paper for the proof of best-approximation results. This and
subordinated $L^2$ error estimates for locally refined triangulations appear of
independent interest. The analysis of optimal convergence rates of an adaptive
mesh-refining algorithm is performed in $3$D and highlights a new version of
discrete reliability.
