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.
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.