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

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

2011-03-29

arxiv.org/abs/1103.5588v1

Physics

Mathematical Physics

21 pages, 5 figures

Scientific paper

A numerical algorithm to solve the spectral problem for arbitrary self-adjoint extensions of 1D regular Schroedinger operators is presented. The construction of all self-adjoint extensions of the symmetric Schroedinger operator on a compact manifold of arbitrary dimension with boundary is discussed. The self-adjoint extensions of such symmetric operators are shown to be in one-to-one correspondence with the group of unitary operators on a Hilbert space of boundary data, refining in this way well-known theorems on the existence of self-adjoint extensions for Laplace-Beltrami operators. The corresponding self-adjoint extensions are characterized by a generalized class of boundary conditions that include the well-known Dirichlet, Neumann, Robin boundary conditions, etc. Only the numerical solution of 1D regular cases are consider in this paper. They constitute however a non-trivial problem. The corresponding numerical algorithms are constructed and their convergence is proved. An appropriate basis of boundary functions must be introduced to deal with arbitrary boundary conditions as described by the general theory. Some significant numerical experiments are also discussed.

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