Let $n\geq 1$. Let $q\in C^{\infty}_0(\mathbb R^n)$ be compactly supported and consider the operator $P= -\Delta+q(x)$ on $\mathbb R^n$. We will assume that $q$ is sufficiently small so that the spectrum of $P$ is purely continuous and given by $[0,\infty)$. Given each $\lambda>0$ and each unit vector $\omega \in \mathbb S^{n-1}$ consider the "generalized eigenfunction" $\phi_{\lambda,\omega}$ as the unique solution to the equation $$ P \phi_{\lambda,\omega} = \lambda^2\, \phi_{\lambda,\omega} \quad \text{on $\mathbb R^n$},$$ subject to some suitable decay condition at infinity for the function $$ \phi_{\lambda,\omega}(x)- e^{i \lambda\omega\cdot x} \quad \text{with $|x|$ large}.$$
Does this give rise to a generalized Fourier transform theory for functions $f\in L^2(\mathbb R^n)$, in the sense of existence of measure $\mu_{\lambda,\omega}$ such that $$ f(x) = \int_{\mathbb R^n} \widehat{f}(\lambda,\omega)\, \phi_{\lambda,\omega}(x)\,d\mu_{\lambda,\omega}$$ and an analogous formula for its inverse transform, possibly with a different measure?
I know that such questions have been answered affirmatively in dimension one, but I suspect this can not be done in higher dimensions?
