Assume $L$ is unbounded, non-self adjoint operator for functions over two space dimensions $(x,y)\in \mathbb{R}^2$, such that upon fourier transforming w.r.t $y$, one can reduce the operator to (for fixed $y$ frequency $\omega$):

$\tilde{L}_{\omega}:=\partial_{xx}+f(x)\partial_x +(i\omega x)$.

Are there any issues in claiming that $spec(L)=\cup_{\omega\in\mathbb{R}} spec(\tilde{L}_{\omega}).$

It seems obvious that if the underlying domain is bounded in $y$ and we work with $y-$periodic functions, then the set of frequencies $\omega$ is just the integers, and above sum is over countable set of frequencies. Hence, I am mostly concerned when we are working with unbounded domain.