Matching Methods

Matching methods construct solutions satisfying separate boundary conditions and join them at an interior point. For a second-order Schrödinger equation, a bound-state energy is found when the left and right solutions become linearly dependent. This approach reduces the unstable traversal of long forbidden regions that can spoil one-sided shooting.

Required background. Shooting methods defines the energy-dependent ODE; physical boundary conditions specifies continuity and derivative jumps.

Wronskian matching

Let $u_L(x;E)$ satisfy the left boundary condition and $u_R(x;E)$ the right one. For a constant-mass equation $u''=q(x,E)u$ with no first derivative, define

$$W(E;x_m)=u_L(x_m)u_R'(x_m)-u_L'(x_m)u_R(x_m).$$

Differentiation gives $\partial_xW=u_Lu_R''-u_L''u_R=0$ wherever both solve the same smooth equation. Thus $W=0$ means the two solution vectors are proportional and can form one global eigenfunction. A numerical drift of the appropriately scaled Wronskian with matching point is a useful integration diagnostic.

Amplitudes can be rescaled independently without changing the zeros of $W$. To avoid comparing dimensionally unlike components, introduce a reference length $\ell$ and vectors $v_L=(u_L,\ell u_L')^T$, $v_R=(u_R,\ell u_R')^T$. A normalized determinant residual is

$$R_W=\frac{\det(v_L,v_R)}{\|v_L\|\|v_R\|}.$$

For a real bound-state equation this gives a scale-independent real residual. Keep scaling choices continuous during a bracketed energy search, or use a subspace-angle method whose root logic is designed for phase changes.

Logarithmic derivatives and their poles

Where both functions are nonzero, the matching condition is equivalent to

$$F(E)=\frac{u_L'}{u_L}-\frac{u_R'}{u_R}=0.$$

The logarithmic derivative $Y=u'/u$ obeys the Riccati equation $Y'=q-Y^2$. It avoids exponentially large amplitudes but develops poles at nodes. A sign change across a pole is not an eigenvalue zero. Vary the matching point, monitor node counts, and confirm a candidate with a Wronskian or reconstructed wavefunction.

For coupled-channel problems, propagate a fundamental solution matrix and its logarithmic derivative or an orthonormalized solution subspace. Forming a long product of raw transfer matrices can destroy the smaller solution directions. The scalar lesson generalizes: stable subspace information matters more than absolute amplitudes.

An eigenstate whose logarithmic derivative has a pole

The free Dirichlet box $0<x<L$ makes the node issue explicit. At positive energy $E=\alpha k^2$, choose

$$u_L(x)=\frac{\sin(kx)}k, \qquad u_R(x)=\frac{\sin[k(L-x)]}k.$$

The right solution has $u_R(L)=0$ and $u_R'(L)=-1$. Their Wronskian is independent of the matching point:

$$W=-\frac{\sin(kL)}k.$$

It vanishes at every physical box level $kL=n\pi$, $n\ge1$. But at $x_m=L/2$ the logarithmic-derivative residual is

$$F(E)=2k\cot(kL/2),$$

which has zeros only for odd $n$ and poles for even $n$. The even-$n$ eigenfunctions have a node at the midpoint, so dividing by their value loses the matching condition exactly where it is needed. This also explains why simply accepting every sign change, or discarding every pole as unrelated to the spectrum, is unreliable. Move the matching point away from the candidate state’s node or use the Wronskian; verify the full boundary solution and its node count.

Finite-well matching example

Consider a symmetric well $V=-V_0$ for $|x|<a$ and zero outside, with $V_0>0$ and a bound energy $-V_0<E<0$. Define

$$k=\sqrt{\frac{E+V_0}{\alpha}}, \qquad \kappa=\sqrt{\frac{-E}{\alpha}}, \qquad \alpha=\frac{\hbar^2}{2m}.$$

For an even state, take $u_L=\cos(kx)$ inside and an exterior decaying solution whose logarithmic derivative at $a$ is $-\kappa$. Matching gives $k\tan(ka)=\kappa$. A pole-free equivalent on this boundary is

$$k\sin(ka)-\kappa\cos(ka)=0.$$

For odd states, $u_L=\sin(kx)$ gives $-k\cot(ka)=\kappa$, or $k\cos(ka)+\kappa\sin(ka)=0$. Search only the physical interval and treat the endpoints separately. Multiplying an equation by a factor that vanishes at a limiting parameter can introduce apparent roots unless the original boundary solution is checked.

The two forms are useful independent tests: the trigonometric residual supplies robust bracketing away from endpoints, while the logarithmic derivative explains the physical matching. The analytic well is a benchmark for a numerical integration/matching implementation, not a requirement that every potential be piecewise constant.

Interfaces and derivative jumps

For a constant-mass delta interaction $g\delta(x-a)$, continuity of $u$ and integration across the singularity give

$$u'(a^+)-u'(a^-)=\frac{g}{\alpha}u(a).$$

The interface acts on $(u,u')^T$ by

$$\begin{pmatrix}u(a^+)\\u'(a^+)\end{pmatrix} = \begin{pmatrix}1&0\\g/\alpha&1\end{pmatrix} \begin{pmatrix}u(a^-)\\u'(a^-)\end{pmatrix}.$$

Incorporate this jump before evaluating a matching residual. Demanding derivative continuity instead solves a different Hamiltonian. For discontinuous effective masses, derive the flux variable from the specified kinetic operator before selecting the interface matrix.

Error control and resonant boundaries

Refine ODE tolerance, starting asymptotic location, and matching position independently. Reconstruct the eigenfunction, normalize it, and verify the residual over both subdomains and the interface. A matched derivative at one point does not establish integration accuracy elsewhere.

Outgoing resonance conditions generally give complex energies and exponentially growing spatial behavior on the real axis. A real-energy sign-change search is then inappropriate. Use the formulations in resonance computation, with their analytic continuation and stability checks.

Exercises

Why is a Wronskian preferable to a logarithmic-derivative residual if the matching point is near a node?

Solution

The logarithmic derivative divides by a nearly zero wavefunction and can develop a pole. The Wronskian uses products of values and derivatives and remains finite for finite solution vectors. Scaling those vectors also prevents large amplitude differences from dominating its numerical evaluation.

References

• Ascher, U. M., R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, 1995 — stable matching and boundary-value conditioning.
• Johnson, B. R., “The Multichannel Log-Derivative Method for Scattering Calculations,” Journal of Computational Physics 13, 445–449 (1973), original article — propagation of logarithmic derivatives.