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Consider the Sturm-Liouville problem

$$-u''(x) + V(x)u(x) = \lambda u(x)$$

for $u : \mathbb R \rightarrow \mathbb R$, $\lambda \in \mathbb R$. I am looking for a method to find the eigenvalues $\lambda$ which produce solutions $u \in L^2(\mathbb R)$ for various choices of $V$, e.g. $V(x) = c/\cosh^2(Cx)$ or $V(x) = e^x$. Is there a general procedure for doing this? If not, can someone provide suggestions for finding the eigenvalues at least in the two cases I mentioned?

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Yes, there is such a procedure. I recall having read something related to this in the classical book on ordinary differential equations by Coddington and Levinson. See here. –  Giuseppe Negro Jan 2 '13 at 18:37

3 Answers 3

up vote 3 down vote accepted

Numerically the problem can be solved e.g. using the Numerov algorithm with the shooting method. There are only a few choices for $V$ for which there is an explicit analytic solution, some of them you can find here.

One of it is $$V(x) = \frac{c}{\cosh^2(C x)}$$ which is called the Pöschl–Teller potential. However, it only has solutions in $L^2$ if $c<0$. After proper rescaling, you will find that the eigensolutions solutions are Legendre functions.

When $V(x) =e^{x}$ I believe that the eigenvalue problem does not have any solution in $L^2$.

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Nice! I wasn't familiar with the Ploeschl-Teller potential. –  Ron Gordon Jan 3 '13 at 0:21

The eigenvalues $\lambda$ emerge from boundary conditions that are part of the differential operator. In your case, $x \in (-\infty, \infty)$, so you may specify the behavior of $u$ as $x \rightarrow \infty$. One way to attack this is to consider the Fourier transform of $u$:

$$ \hat{u}(k) = \int_{-\infty}^{\infty} dx \: u(x) \exp{(i 2 \pi k x)} $$

Applying the integral operator to the S-L equation, and integrating by parts (and assuming that $u$ and its derivative vanish at $\infty$), we get an integral equation for $ \hat{u}$:

$$ \int_{-\infty}^{\infty} dk' \: \hat{V}(k-k') \hat{u}(k') = (\lambda + 4 \pi^2 k^2) \hat{u}(k) $$

where

$$ \hat{V}(k) = \int_{-\infty}^{\infty} dx \: V(x) \exp{(i 2 \pi k x)} $$

and we recover the solution from

$$ u(x) = \int_{-\infty}^{\infty} dk \: \hat{u}(k) \exp{(-i 2 \pi k x)} $$

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How does one solve the integral equation you get after applying the fourier transform? –  user15464 Jan 2 '13 at 18:54
    
You can approximate the convolution by a matrix product. Be warned, however, that the functions $V$ must be $L^2 (\mathbb{R})$. –  Ron Gordon Jan 2 '13 at 18:59
    
Can it be solved explicitly for the choices of $V$ I mentioned in the original post? –  user15464 Jan 2 '13 at 19:26
    
The first, yes, the second, no. But with the second (or anything that is not $L^2(\mathbb{R})$), no technique will work. For those functions, you need to work over a finite interval; in such cases, other techniques may be more appropriate. –  Ron Gordon Jan 2 '13 at 19:28
    
How would I go about solving it for $V(x) = c/\cosh^2(cx)$? –  user15464 Jan 2 '13 at 19:43

There is a technique called "AIM" which stands for "asymptotic iteration method". See for instance this reference 1 and reference 2.

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