  This work deals with the isogeometric Galerkin discretization of the
eigenvalue problem related to the Laplace operator subject to homogeneous
Dirichlet boundary conditions on bounded intervals. This paper uses GLT theory
to study the behavior of the gap of discrete spectra toward the uniform gap
condition needed for the uniform boundary observability/controllability
problems. The analysis refers to a regular $B$-spline basis and concave or
convex reparametrizations. Under suitable assumptions on the reparametrization
transformation, we prove that structure emerges within the distribution of the
eigenvalues once we reframe the problem into GLT-symbol analysis. We also
demonstrate numerically, that the necessary average gap condition proposed in
\cite{bianchi2018spectral} is not equivalent to the uniform gap condition.
However, by improving the result in \cite{bianchi2021analysis} we construct
sufficient criteria that guarantee the uniform gap property.
