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intervoxel
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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
<br /> a_{n+2}+A_n(\lambda)a_{n+1}+B_n(\lambda)a_n=0<br />
where
<br /> A_n(\lambda)=-\frac{(n+1)(2\delta+\epsilon+3(n+\gamma))+Q}{s(n+2)(n+1+\gamma)}<br />
and
<br /> B_n(\lambda)=\frac{(n+\alpha)(n+\beta)}{2(n+2)(n+1+\gamma)}<br />
I know (see http://arxiv.org/abs/hep-th/0207133v2) that the asymptotic behavior of the coefficients is given by
<br /> a_n^{(1)}\sim 2^{-n}n^{-1-\lambda/2}\sum_{s=0}^{\infty}\frac{c_s^{(1)}}{n^s}<br />
and
<br /> a_n^{(2)}\sim n^{-3}\sum_{s=0}^{\infty}\frac{c_s^{(2)}}{n^s}<br />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.
- 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
<br /> a_{n+2}+A_n(\lambda)a_{n+1}+B_n(\lambda)a_n=0<br />
where
<br /> A_n(\lambda)=-\frac{(n+1)(2\delta+\epsilon+3(n+\gamma))+Q}{s(n+2)(n+1+\gamma)}<br />
and
<br /> B_n(\lambda)=\frac{(n+\alpha)(n+\beta)}{2(n+2)(n+1+\gamma)}<br />
I know (see http://arxiv.org/abs/hep-th/0207133v2) that the asymptotic behavior of the coefficients is given by
<br /> a_n^{(1)}\sim 2^{-n}n^{-1-\lambda/2}\sum_{s=0}^{\infty}\frac{c_s^{(1)}}{n^s}<br />
and
<br /> a_n^{(2)}\sim n^{-3}\sum_{s=0}^{\infty}\frac{c_s^{(2)}}{n^s}<br />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.
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