I have been reading that for a wave function to be in the Schwartz function space (highly desirable for physical systems in the Rigged Hilbert Space (RHS) description of QM), among other criteria, it should be representable by a finite, or countably infinite, basis. For instance, a particle in a box, or a harmonic oscillator, can be represented by its countably infinite, orthogonal, energy eigenfunctions as its basis. Note, the wave function for a harmonic oscillator is often described by a Gaussian function, which is relevant to the following question.
A free particle (no potential energy or boundary conditions) can also be described by a Gaussian wave packet (which spreads in space over time). This wave packet is an integral over plane waves, each times a scaling factor which is a function of that plane wave's momentum. In that case, due to the above integral, there is a continuum infinity of energy eigenfunctions; they are not countably infinite. So, the energy eigenfunctions cannot form a countably infinite basis for that Gaussian wave packet. However, I think that the wave packet can still be in Schwartz space by representing it by a countably infinite number of orthogonal Hermite polynomials, which do occupy Schwartz space.
So, my question: is the free particle Gaussian wave packet (with a continuum of energy eigenvalues and eigenfunctions) in the Schwartz space, or in the "dual Schwartz space" of RHS?