  Given a differential operator $\mathcal{L}$ along with its own eigenvalue
problem $\mathcal{L}u = \lambda u$ and an associated algebraic equation
$\mathcal{L}^{(n)} \mathbf{u}_n = \lambda\mathbf{u}_n$ obtained by means of a
discretization scheme (like Finite Differences, Finite Elements, Galerkin
Isogeometric Analysis, etc.), the theory of Generalized Locally Toeplitz (GLT)
sequences serves the purpose to compute the spectral symbol function $\omega$
associated to the discrete operator $\mathcal{L}^{(n)}$
  We prove that the spectral symbol $\omega$ provides a necessary condition for
a discretization scheme in order to uniformly approximate the spectrum of the
original differential operator $\mathcal{L}$. The condition measures how far
the method is from a uniform relative approximation of the spectrum of
$\mathcal{L}$. Moreover, the condition seems to become sufficient if the
discretization method is paired with a suitable (non-uniform) grid and an
increasing refinement of the order of approximation of the method.
  On the other hand, despite the numerical experiments in many recent
literature, we disprove that in general a uniform sampling of the spectral
symbol $\omega$ can provide an accurate relative approximation of the spectrum,
neither of $\mathcal{L}$ nor of the discrete operator $\mathcal{L}^{(n)}$.
