Stationary Scattering Theory

Stationary scattering replaces the long-time limit by boundary values of an energy resolvent. Those boundary values are usually not bounded operators on the original Hilbert space. They become meaningful between weighted spaces, where spatial localization supplies enough regularity to restrict a Fourier transform to an energy shell.

Required background. Resolvent Operators defines the off-spectrum resolvent. Helpful background. Short-Range Potentials states the potential hypotheses; S-Matrix explains energy multiplicity.

The limiting absorption principle changes the operator spaces

Use $H_0=-\Delta$ on $L^2(\mathbb R^d)$, with $\hbar=2m=1$, and define

$$R(z)=(H-z)^{-1},\qquad L^2_s=\{f:\langle x\rangle^s f\in L^2\}.$$

Suppose $V$ is real and $|V(x)|\leq C\langle x\rangle^{-\rho}$ with $\rho>1$, and let $H=H_0+V$ on $H^2$. For every $s>1/2$ and $E>0$, the limits

$$R(E\pm i0)=\lim_{\eta\downarrow0}R(E\pm i\eta)$$

exist in the norm of $\mathcal B(L^2_s,L^2_{-s})$. Equivalently, the sandwiched operators $\langle x\rangle^{-s}R(E\pm i\eta)\langle x\rangle^{-s}$ converge in $\mathcal B(L^2)$.

The limits are locally norm-continuous in positive energy, and are uniformly bounded on compact positive-energy intervals. This is a limiting absorption principle for the stated short-range class. Zero energy is excluded; negative eigenvalues are poles rather than regular boundary points.

The word “absorption” refers to the temporary imaginary part of the spectral parameter. It is a selection procedure for a real self-adjoint problem, not a physical assertion that probability is lost.

Why the unweighted limit cannot exist

For self-adjoint $H$ and $E\in\sigma(H)$,

$$\|R(E+i\eta)\|_{\mathcal B(L^2)}=\frac1\eta.$$

Thus the unweighted operator norms diverge. If the resolvents converged strongly on every $L^2$ vector as $\eta\downarrow0$, uniform boundedness would force bounded norms, a contradiction.

Weighted spaces restrict the sources and enlarge the class of responses. A localized source can generate an outgoing wave that decays spatially but is not square-integrable over all space. The map $L^2_s\to L^2_{-s}$ accommodates precisely this situation.

For compactly supported sources in three dimensions, the free outgoing response has a $1/r$ far field. Its weighted squared norm at infinity behaves like $\int^\infty r^{-2s}dr$, which converges when $s>1/2$. This calculation explains the natural weight threshold in the basic example, without proving the full operator estimate.

Restricting the Fourier transform to an energy sphere

In three dimensions, let $\widehat f$ be the unitary Fourier transform and define

$$\Gamma_0(E)f(\omega) =\frac{E^{1/4}}{\sqrt2}\widehat f(\sqrt E\,\omega), \qquad \omega\in S^2.$$

For arbitrary $f\in L^2$, the right side has no intrinsic pointwise meaning: changing an $L^2$ representative on the sphere does not change its equivalence class. In contrast, $f\in L^2_s$ corresponds to Sobolev regularity of $\widehat f$, and the trace theorem gives

$$\Gamma_0(E):L^2_s(\mathbb R^3)\longrightarrow L^2(S^2) \quad\text{bounded},\qquad s>1/2.$$

The normalization is the one fixed in the energy-shell construction. For sufficiently regular $f,g$,

$$\frac{d}{dE}\langle f,E_{H_0}((-\infty,E])g\rangle =\langle\Gamma_0(E)f,\Gamma_0(E)g\rangle.$$

This is an operator-valued trace statement, not an instruction to evaluate an arbitrary momentum wavefunction pointwise.

How a stationary proof is organized

For the free operator, the Fourier transform and the trace theorem control the spectral density and its regularity. Cauchy-transform estimates then give the weighted free resolvent boundary values.

For the perturbed operator, a resolvent factorization reduces the problem to inversion of an identity plus a compact operator between appropriate weighted spaces. The possible failure of invertibility corresponds to a homogeneous stationary solution satisfying the selected radiation condition. Uniqueness and the absence of positive eigenvalues in the stated potential class remove these obstructions at positive energy.

This is a proof outline, not a derivation of all trace and uniqueness estimates. Yafaev’s Theorem 2.5 supplies the precise result. The next page, Rigorous Lippmann–Schwinger Equation, gives an explicit compact-support Fredholm construction where the spaces and uniqueness argument can be seen directly.

Radiation signs and time labels

With our convention $R(z)=(H-z)^{-1}$, $R_0(E+i0)$ is the outgoing boundary value. The physics resolvent $G_0^{(+)}=(E-H_0+i0)^{-1}$ equals $-R_0(E+i0)$.

A stationary incoming preparation has an incident wave plus outgoing scattered radiation. Its outgoing boundary sign therefore corresponds to the incoming time-limit wave operator $W_-$. Defining the time limit and resolvent convention prevents a sign from carrying two incompatible meanings.

Exercises

1. Show why energy-shell restriction is unbounded on unweighted $L^2$.

Solution

Choose smooth Fourier amplitudes supported in a shell of thickness $\epsilon$ around $|k|=\sqrt E$, with amplitude of order $\epsilon^{-1/2}$ and a fixed angular profile. Their $L^2$ norms stay bounded while their values on the central sphere grow like $\epsilon^{-1/2}$. Weighted source norms penalize the corresponding lack of radial Sobolev control.

2. Does a positive-energy limiting absorption theorem justify setting $E=0$ afterward?

Solution

No. The constants are only uniform on compact subsets of $(0,\infty)$. They may diverge at zero, where a threshold resonance or eigenvalue changes the expansion. A separate zero-energy analysis is required.

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.