R-Matrix Methods

The R-matrix method summarizes the internal scattering region by its response at a boundary, then matches that response to known exterior channels. It is useful when a complicated short-range region can be represented by a finite basis while the long-range problem has analytic or inexpensive numerical solutions. A boundary pole is not automatically a physical resonance pole.

Required background. Matching methods supplies the boundary-value perspective; multichannel scattering defines open and closed channels.

A single-channel boundary convention

Consider a real single-channel potential and a reduced radial function regular at the origin. Choose a boundary radius $a$ and a real dimensionless boundary parameter $b$. Define the dimensionless scalar R matrix through

$$u(a)=R(E)\,[a u'(a)-b u(a)].$$

Thus the exterior logarithmic derivative is $L(E)=[b+R(E)^{-1}]/a$ when these inverses exist. This definition fixes the factors of length and avoids confusing $R$ with $u/u'$.

Introduce a Bloch surface operator so that the internal Hamiltonian plus surface term is Hermitian on the finite interval with the chosen boundary convention. Choose real internal eigenfunctions $u_n$, normalized by $\int_0^a u_n(r)^2\,dr=1$, with eigenvalues $E_n$. The spectral representation is

$$R(E)=\frac{\hbar^2}{2\mu a} \sum_n\frac{u_n(a)^2}{E_n-E}.$$

The surface term restores the boundary contribution from integration by parts. Omitting it while using an arbitrary internal basis changes the boundary problem.

Match without unnecessary inverses

Let the exterior solution be $u=J-NK$, with $K=\tan\delta$ and known radial derivatives. Substitution into the boundary relation gives

$$K= \frac{J-R(aJ'-bJ)} {N-R(aN'-bN)}.$$

This form avoids first computing $R^{-1}$ and can be preferable near a large R-matrix value. Near simultaneous large terms, use scaled linear equations and check conditioning rather than treating the quotient as intrinsically stable.

For channels, $R,J,N,K$ become matrices. The factors do not commute, and the scalar quotient becomes a linear solve with a declared multiplication order. Closed channels must be eliminated or matched to decaying exterior solutions before the open-channel scattering matrix is formed.

A free-region boundary test

For a free $s$ wave, $u(r)=\sin(kr)$, so the exact boundary response is

$$R_{\rm free}(E)= \frac{\sin(ka)} {ka\cos(ka)-b\sin(ka)}.$$

With $b=0$, it is $\tan(ka)/(ka)$. It has poles whenever $\cos(ka)=0$, yet the physical free scattering matrix is still $S=1$ and has no interaction resonance. Correct matching cancels the artificial internal-boundary pole structure.

This is a strong implementation test: converge the internal basis, vary $a$ within the free exterior, and recover zero physical phase across several internal poles. Varying $b$ tests a separate cancellation, which can already be exact in a finite basis.

Boundary-parameter cancellation in a fixed basis

Hold a real finite trial basis $\{\phi_i\}$ fixed as $b$ changes. Write $v_i=\phi_i(a)$, $\alpha=\hbar^2/(2\mu a)$, and let $C_b(E)$ be the matrix of the internal Hamiltonian plus Bloch operator minus $E$ times the overlap matrix. The surface term gives

$$C_b=C_0-\alpha b\,vv^T, \qquad R_b=\alpha v^T C_b^{-1}v.$$

The rank-one inverse identity then yields, wherever the displayed inverses exist,

$$R_b=\frac{R_0}{1-bR_0}, \qquad b+\frac{1}{R_b}=\frac{1}{R_0}.$$

Thus the matched logarithmic derivative and scattering matrix are independent of $b$ at every basis size, up to numerical error. This cancellation is not evidence of basis convergence: the invariant phase can still be inaccurate. Near poles or zeros, use the boundary matching equations rather than subtracting large inverse terms.

If instead the retained space consists of the first $N$ exact internal eigenfunctions for each new $b$, changing $b$ generally changes the trial space itself. A truncated pole sum constructed this way need not obey the fixed-space identity. Distinguish that convergence effect from an inconsistent Bloch term or loss of numerical precision in a fixed-space calculation (Descouvemont and Baye, Section 3.4).

Basis, boundary, and channel convergence

Increase the internal basis at fixed $a$, then vary $a$ within a region where the exterior model is valid. Physical observables should become insensitive to the arbitrary partition. Too small a radius leaves omitted interaction in the exterior; too large a radius can demand many basis functions to resolve free oscillations.

Increase the channel set and closed-channel cutoff independently. A closed channel can affect an open-channel resonance virtually even though it carries no asymptotic flux. Check symmetry of the appropriate real reaction matrix for time-reversal-invariant conventions and unitarity of the flux-normalized open-channel $S$ matrix.

A phenomenological fitted R-matrix parametrization and a calculable internal-basis R matrix use related algebra but have different model and parameter uncertainties. State which problem is being solved.

Exercises

Why can changing the boundary parameter shift internal eigenvalues without changing a scattering phase? Does exact independence of $b$ prove that a finite-basis result is converged?

Solution

In a fixed trial space, the rank-one surface update changes internal eigenvalues and surface residues together. Their response obeys $R_b=R_0/(1-bR_0)$, so the matching combination $b+R_b^{-1}$ is unchanged. For example, $R_0=1/4$ gives $R_1=1/3$ and the same value $4$ for that combination. This identity holds even for an inaccurate one-dimensional trial space, so basis-size and radius convergence still require independent tests. If the truncated space is rebuilt from different boundary-condition eigenfunctions for each $b$, residual dependence can instead reflect the changing approximation space.

References

• Descouvemont, Pierre, and Daniel Baye, “The R-matrix theory,” Reports on Progress in Physics 73, 036301 (2010), arXiv:1001.0678 — internal-region bases and channel matching.