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Physics Letters A

Upper and lower bounds for the Schrödinger-equation eigenvalues

Francisco Marcelo Fernández, John F. Ogilvie, Richard H. Tipping

Published Jul 1, 1986• 6 citations
Publisher
DOI
10.1016/0375-9601(86)90369-5

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The Schrödinger equation in D-dimensions for the Manning-Rosen potential with the centrifugal term is solved approximately to obtain bound states eigensolutions (eigenvalues and eigenfunctions).The Nikiforov-Uvarov (NU) method is used in the calculations.We present numerical calculations of energy eigenvalues to two-and four-dimensional systems for arbitrary quantum numbers n and l with three different values of the potential parameter α.It is shown that because of the interdimensional degeneracy of eigenvalues, we can also reproduce eigenvalues of a upper/lower dimensional sytem from the well-known eigenvalues of a lower/upper dimensional system by means of the transformation (n, l, D) → (n, l ± 1, D ∓ 2)..This solution reduces to the Hulthén potential case.
Abstract The Schrödinger equation in D‐dimensions for the Manning‐Rosen potential with the centrifugal term is solved approximately to obtain bound states eigensolutions (eigenvalues and eigenfunctions). The Nikiforov‐Uvarov (NU) method is used in the calculations. We present numerical calculations of energy eigenvalues to two‐ and four‐dimensional systems for arbitrary quantum numbers n and l with three different values of the potential parameter α. It is shown that because of the interdimensional degeneracy of eigenvalues, we can also reproduce eigenvalues of a upper/lower dimensional system from the well‐known eigenvalues of a lower/upper dimensional system by means of the transformation (n,l,D) → (n,l ± 1,D ∓ 2). This solution reduces to the Hulthén potential case.
A method for systematically improving Hall–Post–Stenschke lower bounds to the bound state eigenvalues of three-body Schrödinger equations is given. The improved bounds are obtained by solving coupled one variable integral equations; the bounds get better as the number of coupled equations is increased. The method generates explicit wave functions which can be used to obtain complementary upper bounds via the Rayleigh–Ritz variational method. Either identical or nonidentical particles can be handled. The method is illustrated by calculations for three identical particles bound by Hooke’s law forces. A brief discussion of extensions to more than three particles is given.