I have trouble to directly obtain the continuous spectrums of differential operators such as $\pm\frac{d}{dx}, \pm i\frac{d}{dx}, \pm \frac{d^2}{dx^2} \pm \frac{d}{dx}$, etc. I find it easier to calculate directly what the point spectrum is via finding eigenvalues, and the residual spectrum via finding the point spectrum of the adjoint. But there seems to be no equivalent method to do that for the continuous spectrum.
One way that I rely on to find the continuous spectrum is to first obtain the inverse of the operator, then shows that it is bounded. Secondly, sometimes the domain is nice enough that I can obtain the result without actually trying.
Is there some method to directly obtain the continuous spectrum without relying on fancy theorems or special cases?
For a specific example, take any of the above operator and use the domain $C^1_0[0,1]$, or $C^1[0,1]$.