15 research outputs found
Generalized spiked harmonic oscillator
A variational and perturbative treatment is provided for a family of
generalized spiked harmonic oscillator Hamiltonians H = -(d/dx)^2 + B x^2 +
A/x^2 + lambda/x^alpha, where B > 0, A >= 0, and alpha and lambda denote two
real positive parameters. The method makes use of the function space spanned by
the solutions |n> of Schroedinger's equation for the potential V(x)= B x^2 +
A/x^2. Compact closed-form expressions are obtained for the matrix elements
, and a first-order perturbation series is derived for the wave
function. The results are given in terms of generalized hypergeometric
functions. It is proved that the series for the wave function is absolutely
convergent for alpha <= 2.Comment: 14 page
Matrix elements for a generalized spiked harmonic oscillator
Closed-form expressions for the singular-potential integrals
are obtained with respect to the Gol'dman and Krivchenkov eigenfunctions for
the singular potential V(x) = B x^2 + A/x^2, B > 0, A >= 0. These formulas are
generalizations of those found earlier by use of the odd solutions of the
Schroedinger equation with the harmonic oscillator potential [Aguilera-Navarro
et al, J. Math. Phys. 31, 99 (1990)].Comment: 12 pages in plain tex with 1 ps figur
Closed-form sums for some perturbation series involving associated Laguerre polynomials
Infinite series sum_{n=1}^infty {(alpha/2)_n / (n n!)}_1F_1(-n, gamma, x^2),
where_1F_1(-n, gamma, x^2)={n!_(gamma)_n}L_n^(gamma-1)(x^2), appear in the
first-order perturbation correction for the wavefunction of the generalized
spiked harmonic oscillator Hamiltonian H = -d^2/dx^2 + B x^2 + A/x^2 +
lambda/x^alpha 0 0, A >= 0. It is proved that the
series is convergent for all x > 0 and 2 gamma > alpha, where gamma = 1 +
(1/2)sqrt(1+4A). Closed-form sums are presented for these series for the cases
alpha = 2, 4, and 6. A general formula for finding the sum for alpha/2 = 2 + m,
m = 0,1,2, ..., in terms of associated Laguerre polynomials, is also provided.Comment: 16 page
Green's function for a Schroedinger operator and some related summation formulas
Summation formulas are obtained for products of associated Lagurre
polynomials by means of the Green's function K for the Hamiltonian H =
-{d^2\over dx^2} + x^2 + Ax^{-2}, A > 0. K is constructed by an application of
a Mercer type theorem that arises in connection with integral equations. The
new approach introduced in this paper may be useful for the construction of
wider classes of generating function.Comment: 14 page
Perturbation expansions for a class of singular potentials
Harrell's modified perturbation theory [Ann. Phys. 105, 379-406 (1977)] is
applied and extended to obtain non-power perturbation expansions for a class of
singular Hamiltonians H = -D^2 + x^2 + A/x^2 + lambda/x^alpha, (A\geq 0, alpha
> 2), known as generalized spiked harmonic oscillators. The perturbation
expansions developed here are valid for small values of the coupling lambda >
0, and they extend the results which Harrell obtained for the spiked harmonic
oscillator A = 0. Formulas for the the excited-states are also developed.Comment: 23 page