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Let $H(f) = -f'' + V(x) f$ be the Schrodinger operator on $\mathbb R$. I am trying to calculate the spectrum (eigenvalues) of the operator $H$ in $L^2(\mathbb R)$ for various choices of $V$. In particular, how does one calculate the spectrum of $H$ if $V(x) = - \frac{C_1}{\cosh^2(C_2 x)}$, $C_1 , C_2 > 0$, or $V(x) = e^x$?

I know that one can find the spectrum by explicitly solving the differential equation $H(f) = Ef$, but I am not sure how to do so.

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You have to solve the time-independent Schroedinger equation,

$$H[f_n](x) = E_n f_n(x)$$

where $E_n$ is an eigenvalue and $f_n(x)$ is the corresponding eigenfunction. Just write out the equation as a plain old differential equation and use the standard techniques you know for solving it. Each value of $E_n$ for which a solution exists is an eigenvalue, and the corresponding solution is the eigenfunction.

If you're doing the analogous problem with multiple variables, you solve it as a partial differential equation instead, i.e. you'll usually have to perform separation of variables first.

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How does one solve the equation $H(f_n) = E_n f_n$? I tried entering this in Wolfram alpha for various values of $E_n$ and it does not seem to give a closed form solution. –  user15464 Dec 9 '12 at 13:28
    
Ah, well now you're asking the question "How do I solve this differential equation?" for a couple of specific differential equations. I'll try to expand on that more soon, when I have time. It would help if you edit your question to make it more clear that you're looking for help solving these specific differential equations. –  David Z Dec 9 '12 at 20:42
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