Point and continuous spectrum of a multiplication operator in $L^2(\mathbb{R})$

Question

Let $M_\alpha f(x) = sin(\frac{x}{\alpha})f(x)$. Compute its spectrum.

My idea is to find the measure of the set $A_\lambda = \left\{x \in \mathbb{R} : sin(\frac{x}{\alpha}) = \lambda\right\}$. If the measure of $A_\lambda$ is zero, then $\lambda$ is not an eigenvalue. So, is it true that in this case the point spectrum is empty? What about the continuous spectrum?