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

Understand

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.