Equivalence with the Automorphic Schwartz Space

Establish whether the projective limit of the Sobolev-like spaces defined by the powers of Sands’ Hamiltonian, \(\varprojlim_\ell V^\ell\), is equivalent to established characterizations of the automorphic Schwartz space, including those of Casselman and Harish-Chandra.

Background

The paper defines Sobolev-like norms using powers of Sands’ Hamiltonian $S$ and forms the projective limit $\varprojlim_\ell V^\ell$. This construction is intended to provide an analogue of the classical characterization of the Schwartz space on $\mathbb{R}$ via the quantum harmonic oscillator.

The authors do not establish that this operator-defined space agrees with existing notions of automorphic Schwartz space. They specifically identify comparison with the Casselman and Harish-Chandra characterizations as unresolved.