Real Cauchy data for a chosen coefficient

For real Cauchy data $(q,p)$, define the linear observable

Its incoming coefficient is

For compact smooth data this follows directly from the canonical field expansion. Its indicated spectral extensions follow by continuity on their moment domains.

Let

It is an antiunitary involution commuting with real functions of $\omega$. It incorporates the reality convention of the incoming scattering modes; componentwise conjugation must not be substituted for it.

Set

and define

These give real $q,p$, satisfy $k(q,p)=k_{-}$, and have spectral support in the compact positive band. They belong to every integer Sobolev space, by spectral functional calculus and elliptic regularity for $\mathsf{L}$.