Scattering States

A scattering state may mean a normalizable vector whose long-time motion resembles a free packet, or a stationary generalized eigenfunction labeled by energy and channel. These meanings are related through wave operators and spectral transforms, but they are not interchangeable. Transporting a generalized ket also needs a continuity statement on test spaces that ordinary Hilbert-space boundedness does not provide.

Required background. Momentum Kets constructs free spectral functionals. Generalized Eigenfunction Expansions explains measure and multiplicity. This page assumes that the wave operators specified below exist; their existence and completeness criteria belong to Mathematical Scattering.

Wave operators transport normalizable packets

Let $H_0$ and $H$ be self-adjoint free and interacting Hamiltonians on the same Hilbert space. The incoming wave operator, when it exists, is

$$W_{\rm in} =\operatorname*{s-lim}_{t\to-\infty} e^{itH/\hbar}e^{-itH_0/\hbar}P_{\rm ac}(H_0).$$

The outgoing operator uses $t\to+\infty$. The limit is strong on each vector, not an operator-norm limit. On the free absolutely continuous subspace, the operator is an isometry.

For an incoming packet $\phi$ in that subspace, $\Psi=W_{\rm in}\phi$ is therefore a normalized interacting vector whenever $\phi$ is normalized. Its defining asymptotic relation is

$$\|e^{-itH/\hbar}\Psi-e^{-itH_0/\hbar}\phi\| \longrightarrow0\quad(t\to-\infty).$$

This is a Hilbert-space statement about packets. It neither requires an exact-energy physical preparation nor claims that the interaction vanishes at every point along a classical trajectory.

A generalized scattering ket needs a transported test space

For a free particle on the line, write a free packet through its momentum amplitude $a(p)$. Formally one would like to write $|p,\mathrm{in}\rangle=W_{\rm in}|p\rangle$. A bounded operator on $L^2$ does not automatically act on the larger antidual, so this notation needs justification.

Choose an interacting test space $\Phi_H$ such that

$$W_{\rm in}^*:\Phi_H\longrightarrow\mathcal S(\mathbb R)$$

is continuous in the stated test topologies. Define

$$\langle\chi|p,\mathrm{in}\rangle =\langle W_{\rm in}^*\chi|p\rangle, \qquad \chi\in\Phi_H.$$

The right-hand side is now a continuous antilinear functional of $\chi$. This is the precise transported ket. Constructing such a test space, or constructing an equivalent distorted Fourier transform directly, is part of the model’s analysis.

If $H$ preserves $\Phi_H$ continuously and the domain-level intertwining with $H_0$ is valid there, the free generalized eigen-equation implies

$$H^\times|p,\mathrm{in}\rangle =\frac{p^2}{2m}|p,\mathrm{in}\rangle.$$

The spectral intertwining and the test-space extension are separate ingredients. Merely solving the same differential equation does not identify this incoming functional or its normalization.

Packet expansions identify the scattering subspace

For suitable amplitudes and testing vectors, the transported expansion reads

$$\langle\chi,W_{\rm in}\phi\rangle =\int\langle\chi|p,\mathrm{in}\rangle a(p)\,dp.$$

For arbitrary square-integrable amplitudes, reconstruct the free packet in Hilbert norm and apply the bounded wave operator. This gives the Hilbert extension without assuming pointwise convergence of a stationary eigenfunction integral.

The wave-operator range lies in the interacting absolutely continuous subspace. Asymptotic completeness is the stronger assertion that it equals that whole subspace. If completeness is not proved, a stationary expansion over the constructed scattering channels describes only the established range.

Even with completeness on the absolutely continuous subspace, bound states require their separate spectral terms. A possible singular continuous subspace also cannot be discarded without a theorem excluding it. Continuous-spectrum notation alone does not establish the absence of that sector.

Incoming labels and stationary boundary signs need definitions

An incoming packet is defined by its behavior as time tends to minus infinity. A stationary solution can instead be labeled by an outgoing or incoming resolvent boundary condition. Different authors attach plus and minus signs to these constructions differently.

Specify the time limit or resolvent boundary value instead of inferring the meaning from a superscript. Energy alone can also leave channel multiplicity unresolved: in one dimension the free energy has two momentum directions, while higher-dimensional scattering has angular or channel labels. Those labels are needed to reconstruct the state.

Resonant states add another distinction. A pole of an analytically continued resolvent is not a normalizable eigenvector of a self-adjoint Hamiltonian with complex energy. Any generalized resonance expansion needs its own continuation, test space, and remainder control.

Exercise: can a scattering expansion contain a bound component?

Suppose $W_{\rm in}$ exists and $\Psi=W_{\rm in}\phi$ with $\phi$ in the free absolutely continuous subspace. Can $\Psi$ have a nonzero projection onto an interacting bound eigenvector, including an embedded eigenvector at positive energy?

Solution

No. Spectral intertwining preserves the packet’s absolutely continuous spectral measure, so $\Psi\in\mathcal H_{\rm ac}(H)$, orthogonal to the pure point subspace. This excludes embedded as well as isolated bound eigenvectors. An argument using only the sign of free energy would exclude negative-energy bound states but would not address an embedded positive-energy eigenvalue. Completeness is unnecessary for this exclusion; it is needed to claim that every interacting absolutely continuous state arises this way.

References

• de la Madrid, Rafael, “The role of the rigged Hilbert space in quantum mechanics,” European Journal of Physics 26, 287–312 (2005), doi:10.1088/0143-0807/26/2/008, arXiv:quant-ph/0502053 — explicit scattering riggings and generalized eigenfunctionals.
• Teschl, Gerald, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, author’s edition — wave operators, spectral intertwining, and asymptotic completeness.