Eigensolutions of the Wigner–Eisenbud problem for a cylindrical nanowire within finite volume method

We present a node-centered finite volume method for computing a representative range of eigenvalues and eigenvectors of the Schrِdinger operator on a three-dimensional cylindrically symmetric bounded domain with mixed boundary conditions. The three-dimensional Schrِdinger operator is reduced to a family of two-dimensional Schrِdinger operators distinguished by a centrifugal potential. We consider a uniform, boundary conforming Delaunay mesh, which additionally conforms to the material interfaces. We study how the anisotropy of the effective mass tensor acts on the uniform approximation of the first K eigenvalues and eigenvectors and their sequential arrangement. There exists an optimal uniform Delaunay discretization with matching anisotropy with respect to the effective masses of the host material. For a centrifugal potential one retrieves the theoretically established first-order convergence, while second-order convergence is recovered only on uniform grids with an anisotropy correction.