  A numerical algorithm to solve the spectral problem for arbitrary
self-adjoint extensions of 1D regular Schroedinger operators is presented. It
is shown that the set of all self-adjoint extensions of 1D regular Schroedinger
operators is in one-to-one correspondence with the group of unitary operators
on the finite dimensional Hilbert space of boundary data, and they are
characterized by a generalized class of boundary conditions that include the
well-known Dirichlet, Neumann, Robin, (quasi-)periodic boundary conditions,
etc. The numerical algorithm is based on a nonlocal boundary extension of the
finite element method and their convergence is proved. An appropriate basis of
boundary functions must be introduced to deal with arbitrary boundary
conditions and the conditioning of its computation is analyzed. Some
significant numerical experiments are also discussed as well as the comparison
with some standard algorithms. In particular it is shown that appropriate
perturbations of standard boundary conditions for the free particle leads to
the theoretically predicted result of very large absolute values of the
negative groundlevels of the system as well as the localization of the
corresponding eigenvectors at the boundary (edge states).
