This is a cross-post from: https://math.stackexchange.com/questions/4651664/representation-of-an-operator-on-a-generalized-eigenfunction
Suppose we have an (essentially) self adjoint operator $L$ acting on Schwartz functions with a Rigged Hilbert space $S\subset L^2(\mathbb{R}) \subset S^*$.
By Gel'fand Maurin theorem $L$ should have a complete set of generalized eigenfunctions in $S^*$. Furthermore, suppose we know that $L$ commutes with $\Delta=\frac{d^2}{dx^2}$, $\Delta L\phi(x)=L\Delta\phi(x)$, so that the FT diagonalizes $L$. In this case $\Delta$ and $L$ should share a common set of generalized eigenfunctions.
Furthermore, we have a representation for the action of $L$ as a countably infinite summation over some basis elements $\chi$ (not necessarily orthogonal): $$L\phi(x)=\sum_n \omega_n\chi_n$$, where $\omega$'s are weights uniquely determined for $\phi\in S$ (by some inner product integral) and $\chi \notin S$.
Question: assuming the representation for $L$, as a summation over $\chi$, was derived for Schwartz functions (for example, convergence of $\omega$s was established for $\phi\in S$). In general, because we know that $L\varphi=\lambda\varphi$ for a generalized eigenfunction $\varphi$, can we run $\varphi$ through the summation representation of $L$ (including the ip used to extract $\omega$s), or somehow transform the basis elements, $\chi$s such that the inner product for the coefficients converges (perhaps distributionally)?
Motivating Example, because we know $e^{i\xi x}$ are generalized eigenfunctions for $\Delta$ (which implies that they are also generalized eigenfunctions for $L$) we should have that $\omega=\int e^{i\xi x}\chi_n dx$ exists, perhaps as a $\delta$ in the case that $\chi=e^{ix\zeta}$.
