Wave, scattering and channel operators

Assume the Axiom of Choice. Let ${\mathcal{H}}_0,\mathcal{H}$ be separable complex Hilbert spaces, $H_0$ and $H$ self-adjoint operators on them, and $J:{\mathcal{H}}_0\to \mathcal{H}$ a bounded linear map. Write $E_{H_0}$ and $E_H$ for their spectral projection valued measures. The absolutely continuous subspace ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H_0)$ consists of vectors $u$ for which the scalar measure $B\mapsto \mathrm{\parallel }{E}_{H_0}(B)u{\mathrm{\parallel }}^2$ is absolutely continuous with respect to Lebesgue measure; $P_{\mathrm{a}\mathrm{c}}(H_0)$ is its orthogonal projection. Define the two-space wave operators, when the pointwise strong limits exist on ${\mathcal{H}}_0$, by

${\mathrm{\Omega }}_{\pm }(H,H_0;J):={\mathrm{s-lim}}_{t\to \pm \mathrm{\infty }}{e}^{itH}J{e}^{-it{H}_0}{P}_{\mathrm{a}\mathrm{c}}(H_0),$

where a strong limit means that for every $u\in {\mathcal{H}}_0$ the vector-valued expression converges in $\mathcal{H}$. The two signs are separate limits. When $J=I$ and ${\mathcal{H}}_0=\mathcal{H}$, write ${\mathrm{\Omega }}_{\pm }(H,H_0)$. A wave operator is complete when its range is all of ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H)$. When both wave operators exist and are isometries on ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H_0)$, the scattering operator on that subspace is

$S(H,H_0;J):={\mathrm{\Omega }}_{+}^{\ast }{\mathrm{\Omega }}_{-}\mathrm{.}$

Whenever both wave operators exist, are isometries on ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H_0)$, and are complete with range ${\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H)$, the scattering operator $S(H,H_0;J)$ is unitary. For a fixed integer $r\ge 1$ and an open energy interval $I\subset \mathbb{R}$, finite-channel data consists of the standard space ${\mathbb{C}}^r$ and a matrix field $S:I\to U(r)$, where

$U(r):=\{U\in M_r(\mathbb{C}):U^{\ast }U=U{U}^{\ast }=I_r\}\mathrm{.}$

Here $U^{\ast }$ is conjugate transpose, $\mathrm{tr}U={\sum }_{j=1}^r{U}_{jj}$, and $\det U$ is the usual alternating determinant. This matrix field is separate specified data; the definition does not assert that it is obtained from the abstract wave operators above.

The upper arrow is ${\mathrm{\Omega }}_{-}$ restricted to the free absolutely continuous subspace. The right arrow is the full adjoint ${\mathrm{\Omega }}_{+}^{\ast }:\mathcal{H}\to {\mathcal{H}}_{\mathrm{a}\mathrm{c}}(H_0)$; its codomain restriction follows from the factor $P_{\mathrm{a}\mathrm{c}}(H_0)$ in the definition of ${\mathrm{\Omega }}_{+}$. Whenever both limits exist, the diagonal arrow is their composite; if both are also isometries, this is the scattering operator $S={\mathrm{\Omega }}_{+}^{\ast }{\mathrm{\Omega }}_{-}$ defined above. The diagram records this composition only and asserts neither existence nor completeness.

The Axiom of Choice

A bounded linear operator between normed spaces

Symmetric, self-adjoint and essentially self-adjoint operators

A self-adjoint operator generates a strongly continuous unitary group

Canonical decomposition into pure point, absolutely continuous and singular continuous parts