I have the following problem:
Determine the point spectrum and the continuous spectrum of the operator $$(A\psi )(x)=\theta (x)(\cos x)\psi (x)$$ on $L_2(\mathbb R,dx)$, where $\theta(x)=0$ for $x<0$, $\theta(x)=1$ for $x\geq 0$.
Now, for the point spectrum I guess I only have to find a $\psi(x)$ such that $(A\psi )(x)=\theta(x)(\cos x)\psi(x)=\lambda\psi (x)$ for some $\lambda\in\mathbb C$, but I don't know where to look. Or even if this is the way of dealing with problems like these. As for the continuous spectrum, I don't really know where to start.
Basically I've been trying to "guess" a good $\psi$ and looking for a clue in the Fourier transform/Fourier series of the equation, but I haven't come up with anything. I've only really dealt with eigenvalues/eigenvectors in the context of matrices and differential equations before.
Also, if anyone knows of any internet resources with problems like these (and preferably a few examples of how to solve them,) I'd be happy to know.