  We extend the conforming virtual element method to the numerical resolution
of eigenvalue problems with potential terms on a polytopal mesh. An important
application is that of the Schrodinger equation with a pseudopotential term.
This model is a fundamental element in the numerical resolution of more complex
problems from the Density Functional Theory. The VEM is based on the
construction of the discrete bilinear forms of the variational formulation
through certain polynomial projection operators that are directly computable
from the degrees of freedom. The method shows a great flexibility with respect
to the meshes and provide a correct spectral approximation with optimal
convergence rates. This point is discussed from both the theoretical and the
numerical viewpoint. The performance of the method is numerically investigated
by solving the Quantum Harmonic Oscillator problem with the harmonic potential
and a singular eigenvalue problem with zero potential for the first
eigenvalues.
