  We show that isogeometric Galerkin discretizations of eigenvalue problems
related to the Laplace operator subject to any standard type of homogeneous
boundary conditions have no outliers in certain optimal spline subspaces.
Roughly speaking, these optimal subspaces are obtained from the full spline
space defined on certain uniform knot sequences by imposing specific additional
boundary conditions. The spline subspaces of interest have been introduced in
the literature some years ago when proving their optimality with respect to
Kolmogorov $n$-widths in $L^2$-norm for some function classes. The
eigenfunctions of the Laplacian -- with any standard type of homogeneous
boundary conditions -- belong to such classes. Here we complete the analysis of
the approximation properties of these optimal spline subspaces. In particular,
we provide explicit $L^2$ and $H^1$ error estimates with full approximation
order for Ritz projectors in the univariate and in the multivariate
tensor-product setting. Besides their intrinsic interest, these estimates imply
that, for a fixed number of degrees of freedom, all the eigenfunctions and the
corresponding eigenvalues are well approximated, without loss of accuracy in
the whole spectrum when compared to the full spline space. Moreover, there are
no spurious values in the approximated spectrum. In other words, the considered
subspaces provide accurate outlier-free discretizations in the univariate and
in the multivariate tensor-product case. This main contribution is complemented
by an explicit construction of B-spline-like bases for the considered spline
subspaces. The role of such spaces as accurate discretization spaces for
addressing general problems with non-homogeneous boundary behavior is discussed
as well.
