Asymptotic behavior of coefficients

[intervoxel]

The eigenvalue problem of Schroedinger equation can be solved in a variety of ways. The continued fraction method can be stated by the following recipe:

- represent the solution of the D.E. as a power series

- replace back this solution into the D.E.

- obtain a three term recurrence relation by aligning its indices;

- obtain the asymptotic behavior of the two coefficients -- the general asymptotic behavior is a combination of these terms

- check if the limit of the ratio of these coefficients as n tends to infinity is zero

- case positive, the solution used in the numerator is the minimal solution

- the quantization condition is given by c_1(lambda)=0;

- calculate the modulus of the ratio between successive terms of these solutions

- check the series convergence using these values (make a series expansion if necessary);

- if it converges then we can invoke Pincherle's theorem to create a continued fraction expansion and thereby a transcendental equation for finding the eigenvalues.

My question is:

Given the difference equation

$$a_{n+2}+A_n(\lambda)a_{n+1}+B_n(\lambda)a_n=0$$

where

$$A_n(\lambda)=-\frac{(n+1)(2\delta+\epsilon+3(n+\gamma))+Q}{s(n+2)(n+1+\gamma)}$$

and

$$B_n(\lambda)=\frac{(n+\alpha)(n+\beta)}{2(n+2)(n+1+\gamma)}$$

I know (see http://arxiv.org/abs/hep-th/0207133v2) that the asymptotic behavior of the coefficients is given by

$$a_n^{(1)}\sim 2^{-n}n^{-1-\lambda/2}\sum_{s=0}^{\infty}\frac{c_s^{(1)}}{n^s}$$

and

$$a_n^{(2)}\sim n^{-3}\sum_{s=0}^{\infty}\frac{c_s^{(2)}}{n^s}$$I have to do the a similar calculation in my research project but I couldn't find out the procedure used to calculate the asymptotic expansion above. Please, can someone show me the steps to such a solution?

I tried to follow the text by Saber Elaydi, An Introduction to Difference Equations without success.

 

[intervoxel]

Oh, come on. Could anyone at least cite a good book about this subject, please?

 

[Avodyne]

- obtain a three term recurrence relation by aligning its indices;

You only get a 3-term relation in very special cases, such as the harmonic oscillator and the hydrogen atom. So, for most problems, this method doesn't work.

Did you try the reference cited in the hepth paper you mention?

R. Wong and H. Li, “Asymptotic expansions for second order linear difference equations”, J. Comput. Appl. Math. 41 (1992) 65.