Wave Operators

Wave operators turn asymptotic comparison into a bounded operator between spectral subspaces. Once their strong limits exist, isometry, spectral intertwining, and exclusion of bound components follow without a potential-specific calculation. Those structural conclusions are proved here; Cook’s Method and Short-Range Potentials supply separate existence criteria.

Helpful background. Free and Interacting Dynamics introduces the comparison operator. Spectral Theorem for Unbounded Operators supplies spectral measures and the functional calculus.

Strong comparison limits on the absolutely continuous subspace

Let $H_0,H$ be self-adjoint on $\mathcal H$, and put $P_0=P_{\rm ac}(H_0)$. Define, when the limits exist on every vector,

$$W_\pm=\operatorname*{s-lim}_{t\to\pm\infty} e^{itH/\hbar}e^{-itH_0/\hbar}P_0.$$

Here the subscript is the sign of time: $W_-=W_{\rm in}$ and $W_+=W_{\rm out}$. The physics convention $\Omega^{(+)}$ for an incoming state therefore corresponds to $W_-$, not $W_+$. A boundary-condition superscript and a time-limit subscript need not match.

The projection $P_0$ removes the free singular spectral sectors. For the ordinary free Laplacian on $\mathbb R^d$, $P_0=I$. For an abstract reference Hamiltonian it need not be the identity.

Existence on a dense subset of $P_0\mathcal H$ suffices. Indeed, each comparison operator has norm at most one. Approximating an arbitrary vector by a dense-set vector bounds the difference of two late-time comparison vectors by twice the approximation error plus a term already known to be Cauchy. Completeness of $\mathcal H$ then supplies the limit.

Isometry gives a closed scattering range

For every $\phi$,

$$\|W_\pm\phi\| =\lim_{t\to\pm\infty}\|e^{itH/\hbar}e^{-itH_0/\hbar}P_0\phi\| =\|P_0\phi\|.$$

Polarization therefore gives

$$W_\pm^*W_\pm=P_0.$$

Thus $W_\pm$ vanishes on $P_0^\perp\mathcal H$ and is an isometry on $P_0\mathcal H$. Its range $\mathcal R_\pm$ is closed: a convergent sequence $W_\pm\phi_n$ makes $\phi_n$ Cauchy in the initial subspace. The operator $W_\pm W_\pm^*$ is the orthogonal projection onto this range.

This is not yet the equality $W_\pm W_\pm^*=P_{\rm ac}(H)$. That equality is the additional completeness assertion.

Intertwining follows by shifting the time limit

Write $U(t)=e^{-itH/\hbar}$ and $U_0(t)=e^{-itH_0/\hbar}$. For fixed real $s$,

$$U(s)U(t)^*U_0(t) =U(t-s)^*U_0(t-s)U_0(s).$$

Taking the strong limit, and using that $P_0$ commutes with $U_0(s)$, gives

$$U(s)W_\pm=W_\pm U_0(s).$$

The same relation with $-s$ shows that $\mathcal R_\pm$ is invariant under both $U(s)$ and its inverse. Thus it reduces the unitary group and its generator.

Uniqueness of spectral measures, or equivalently the bounded Borel functional calculus, now yields

$$E_H(B)W_\pm=W_\pm E_{H_0}(B)$$

for every Borel set $B$. For $\phi\in P_0\mathcal H$,

$$\langle W_\pm\phi,E_H(B)W_\pm\phi\rangle =\langle\phi,E_{H_0}(B)\phi\rangle.$$

The scalar spectral measure is preserved exactly, so $\mathcal R_\pm\subseteq\mathcal H_{\rm ac}(H)$. In particular, no wave-operator packet has an interacting eigenvector component, even at an embedded eigenvalue.

Domain intertwining is a consequence, not a formal product

The preserved spectral measure also preserves its second moment. Consequently

$$W_\pm\bigl(D(H_0)\cap P_0\mathcal H\bigr) =D(H)\cap\mathcal R_\pm,$$

and on that domain,

$$HW_\pm\phi=W_\pm H_0\phi.$$

This follows by spectral truncation or differentiation of the group identity. It specifies exactly where the unbounded products are defined. It does not assert $W_\pm D(H_0)=D(H)$, which would improperly include bound or missing continuous states.

The same reasoning applies to measurable functions of energy whenever the corresponding spectral integrability condition holds. Thus a comparison packet’s energy distribution is transported, even though position or momentum distributions generally change during scattering.

The inverse exists on the established range

For $\Psi=W_\pm\phi$ with $\phi\in P_0\mathcal H$, unitarity gives

$$\|U_0(t)^*U(t)\Psi-\phi\| =\|\Psi-U(t)^*U_0(t)\phi\|\longrightarrow0.$$

The inverse comparison therefore converges on $\mathcal R_\pm$, with value $W_\pm^*\Psi$. Extending this inverse limit to every vector of $\mathcal H_{\rm ac}(H)$ is exactly the missing exhaustion issue, treated in Asymptotic Completeness.

Generalized plane waves require a separate extension to test spaces. Scattering States makes that extension explicit; none of the Hilbert-space limits above automatically acts on an arbitrary distribution.

Exercises

1. Show that a wave-operator packet is orthogonal to every eigenvector of $H$.

Solution

For any eigenvalue $\lambda$, spectral intertwining gives $E_H(\{\lambda\})W_\pm\phi=W_\pm E_{H_0}(\{\lambda\})P_0\phi=0$. An absolutely continuous spectral measure assigns zero mass to a singleton. This argument does not require $\lambda$ to lie outside the essential spectrum.

2. If $H_0$ has purely discrete spectrum, what does the definition above produce?

Solution

It gives $P_0=0$ and $W_\pm=0$. This vacuous comparison says nothing about the interacting continuous spectrum. A reference Hamiltonian must contain the asymptotic channels one intends to describe.

References

• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Chapter 12, author’s text.
• D. Yafaev, “Lectures on scattering theory,” 2004, arXiv:math/0403213.