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

Numerical Solutions of the spectral problem for arbitrary self-adjoint extensions of the 1D Schroedinger equation