An asymptotic interation method for solving second-order homogeneous linear differential equations of the form y ′′ = λ 0 (x)y ′ + s 0 (x)y is introduced, where λ 0 (x) = 0 and s 0 (x) are C ∞ functions. Applications to Schrödinger type problems, including some with highly singular potentials, are presented.

We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue problems which includes Schrödinger problems with Coulomb, harmonic oscillator, or Pöschl-Teller potentials, as well as the special eigenproblems studied recently by Bender et al [J. Phys. A: Math. Gen. 34 9835 (2001)] and generalized in the present paper to higher dimensions.

In a previous paper (J. Phys. A 36, 11807 (2003)), we introduced the `asymptotic iteration method' for solving second-order homogeneous linear differential equations. In this paper, we study perturbed problems in quantum mechanics and we use the method to find the coefficients in the perturbation series for the eigenvalues and eigenfunctions directly, without first solving the unperturbed problem.Comment: 13 page

We consider a single particle which is bound by a central potential and obeys the Dirac equation in d dimensions. We first apply the asymptotic iteration method to recover the known exact solutions for the pure Coulomb case. For a screened-Coulomb potential and for a Coulomb plus linear potential with linear scalar confinement, the method is used to obtain accurate approximate solutions for both eigenvalues and wave functions.Comment: 18 page

Conditions for the second-order linear differential equationto have polynomial solutions are given. Several application of these results to Schrödinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis.

We consider the differential equations y ′′ = λ 0 (x)y ′ + s 0 (x)y, where λ 0 (x), s 0 (x) are C ∞ −functions. We prove (i) if the differential equation, has a polynomial solution of degree n > 0, then0 and δ n = 0, then the differential equation has a polynomial solution of degree at most n. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kind), Gegenbauer, and the Hypergeometric type, etc, obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations.

An approximate solution of the Klein-Gordon equation for the general Hulthén-type potentials in D-dimensions within the framework of an approximation to the centrifugal term is obtained. The bound state energy eigenvalues and the normalized eigenfunctions are obtained in terms of hypergeometric polynomials.

A complete variational treatment is provided for a family of spiked-harmonicA compact topological proof is presented that the set S = {ψ n } of known exact solutions for α = 2 constitutes an orthonormal basis of the Hilbert space L 2 (0, ∞) .Closed-form expressions are derived for the matrix elements of H with respect to S . These analytical results, and the inclusion of a further free parameter, facilitate optimized variational estimation of the eigenvalues of H to high accuracy.PACS 03.65.Ge Spiked harmonic oscillators . . .