Numerical Solutions of the Spectral Problem for Arbitrary Self-Adjoint Extensions of the One-Dimensional Schrödinger Equation

Abstract

A numerical algorithm to solve the spectral problem for arbitrary self-adjoint extensions of one-dimensional regular Schrödinger operators is presented. It is shown that the set of all self-adjoint extensions of one-dimensional regular Schrödinger 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, and (quasi-)periodic boundary conditions. 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 lead 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). Introduction. The study of the self-adjointness of Schrödinger operators hasbeen a fundamental mathematical problem since the beginning of quantum mechanics and there is a vast literature on the subject. (See, for instance, [Re75], the review [Si00], and references therein.) In spite of this, there is a continuous flow of new results and even surprises. (See, for instance, the recent papers where some apparent paradoxical aspects of the spectrum of certain self-adjoint extensions of the Schrödinger operator in two dimensions are analyzed [Be08], [Ma09], [Be09].)Consider the evolution of a quantum system on a D-dimensional Riemannian manifold M with boundary ∂M under the influence of a potential V which is given by the Schrödinger equation i ∂Ψ ∂t = HΨ, with H the Hamiltonian operator of the system given by

On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics

“…If we determine the set of self-adjoint extensions of system A, that we denote by M A , shouldn't the set of self-adjoint extensions of the composite system M AB be such that M AB = M A ? It is well known that in this case we have that M A = U(1) , see for instance [AIM05,ILPP13,IPP13,Koc75,BGP08]. However, as we will see by means of a computation later, the set of self-adjoint extensions of the composite system is much bigger.…”

“…In the recent paper [IPP13], the ideas discussed in this lecture have been applied successfully to study the spectrum of Schrödinger operators in 1D and its stable and accurate numerical computation.…”

Edge states: topological insulators, superconductors and QCD chiral bags

“…At the same time, the corresponding eigenstates get progressively more localised at the edge and eventually become weakly zero as µ → ∞. Numerical evidence of this phenomenon in one dimension and a numerical algorithm to solve such eigenvalue problems can be found in [3].…”

Three lectures on global boundary conditions and the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on Riemannian manifolds with boundary

“…Some of the material presented here has already appeared published elsewhere (see for instance [5] where some of the preliminary ideas on the global topology of the space of self-adjoint extensions for the covariant Laplacian and its relation to topology change appeared for the first time), or will appear in various forms (see for instance [25] for a detailed discussion of 1D Schrödinger operators). The general theory of self-adjoint extensions from the point of view of quadratic forms is discussed in [26] and will not be considered here as well as the theory of self-adjoint extensions with symmetry that will be discussed elsewhere.…”