Smooth vectors of the Schrödinger representation

Statement

For the Schrödinger representation of the real Heisenberg group on $H=L^2(\mathbb R)$ , the smooth vectors are precisely

H^\infty=\mathcal S(\mathbb R),

where $L^2$ classes are identified with their smooth representatives. The smooth-vector Fréchet topology equals the usual Schwartz topology.

Why both regularity and decay appear

The infinitesimal translation and modulation operators are $d/dx$ and $ix$ . Their ordered products require all weighted weak derivatives $x^a f^{(b)}$ to belong to $L^2$ . The iterated-domain characterization therefore tests both spatial differentiability and decay.

Schwartz bounds imply these $L^2$ bounds by adding a sufficiently high polynomial weight. Conversely, applying one-dimensional Sobolev bounds to each $x^a f^{(b)}$ makes it bounded, since it and its derivative lie in $L^2$ . These estimates in both directions identify both the space and its topology. The same statement holds on $\mathbb R^n$ , with multi-indices and sufficiently high Sobolev order.

Pointwise smoothness is not enough

The function

f(x)=(1+x^2)^{-1/2}

is smooth and square-integrable, but $xf\notin L^2$ . It is not in the domain of the modulation generator, so it is not a smooth vector for the Heisenberg representation. Smoothness of a representative function and smoothness of its Hilbert-valued orbit map are different requirements.

The resulting rigging

The space of distribution vectors is the continuous anti-dual $\mathcal S(\mathbb R)^\times$ . Thus

\mathcal S(\mathbb R)\hookrightarrow L^2(\mathbb R) \hookrightarrow\mathcal S(\mathbb R)^\times.

It is the anti-linear version of the familiar tempered-distribution rigging. Unlike an arbitrary smooth-vector space, Schwartz space is nuclear.

References

Rahul Garg and Sundaram Thangavelu, On the structure of analytic vectors for the Schrödinger representation. §2, opening identification of smooth vectors with Schwartz space, and Theorem 2.2. The equivalence of topologies follows from the weighted Sobolev estimates above.