  We discuss the $p$- and the $hp$-versions of the virtual element method for
the approximation of eigenpairs of elliptic operators with a potential term on
polygonal meshes. An application of this model is provided by the Schr\"odinger
equation with a pseudo-potential term. We present in details the analysis of
the p-version of the method, proving exponential convergence in the case of
analytic eigenfunctions. The theoretical results are supplied with a wide set
of experiments. We also show numerically that, in the case of eigenfunctions
with finite Sobolev regularity, an exponential approximation of the eigenvalues
in terms of the cubic root of the number of degrees of freedom can be obtained
by employing $hp$-refinements. Importantly, the geometric flexibility of
polygonal meshes is exploited in the construction of the $hp$-spaces.
