  In this paper we investigate the spectral problem in Finsler geometry. Due to
the nonlinearity of the Finsler-Laplacian operator, we introduce
\textit{faithful dimension pairs} by means of which the spectrum of a compact
reversible Finsler metric measure manifold is defined. Various upper and lower
bounds of such eigenvalues are provided in the spirit of Cheng, Buser and
Gromov, which extend in several aspects the results of Hassannezhad, Kokarev
and Polterovich. Moreover, we construct several faithful dimension pairs based
on Lusternik-Schnirelmann category, Krasnoselskii genus and essential
dimension, respectively; however, we also show that the Lebesgue covering
dimension pair is not faithful. As an application, we show that the
Bakry-\'Emery spectrum of a closed weighted Riemannian manifold can be
characterized by the faithful Lusternik-Schnirelmann dimension pair.
