The Laplacian, $\nabla ^2 = \frac{\partial^2}{\partial x^2 }+ \frac{\partial^2}{\partial y^2 }$, in 2D cartesian coordinates, has eigenfunctions of the form $Ae^{-i(k_x x + k_y y)} + Be^{i(k_x x + k_y y)}$, which is the sum of a pair of plane waves. Each plane wave is defined in all space, with wavefronts from $\pm\infty$ to $\mp\infty$, in the $(k_x, k_y)$ direction. The purely radial version (without $\theta$ or $\phi$ angles) of the Laplacian, $\frac{\partial^2}{\partial r^2} + r^{-1} \frac{\partial}{\partial r}$, has eigenfunctions of the form $A r^{-1}e^{-i(kr)} + B r^{-1}e^{i(kr)}$, which is the sum of a pair of radial waves, where $r=\sqrt{x^2 + y^2}$, in cartesian coordinates. I know that a radial wave can be expanded into a superposition of plane waves by the Weyl Identity. However, those plane waves have different values for their $k_i$s. Also, a plane wave can be expanded into a superposition of radial waves by the Fresnel Kirchhoff formula. Those radial waves all have different center points, which are all on the same plane wavefront, but are “zeroed-out” “behind” the wavefront by the “obliquity factor”. Hence, those plane waves are not defined in all space, in this formula.
Are the above eigenfunctions of the radial Laplacian also eigenfunctions of the cartesian Laplacian, and vice-versa? It seems that the answer is no, but just want to confirm.