Linear independence and error control in Dirac's non-orthogonal mixed-basis first-order perturbation scheme for rearrangement collisions

In a 1955 note, P. A. M. Dirac proposed a first-order perturbation scheme for collision problems in which the interaction energy changes identity during the collision (rearrangement, e.g. charge transfer $A + B^{+} \to A^{+} + B$): the source term is expanded in the union of two mutually non-orthogonal eigenfunction sets belonging to two different partitions of the Hamiltonian, and the expansion coefficients follow from coupled integral equations $(1+\Omega)Y = -Z$. The scheme was left unvalidated: no proof that the union of the two sets is linearly independent (equivalently, that $1+\Omega$ is invertible) in concrete collision problems; no controlled prescription for restricting the basis when independence fails; no error bounds for discarding the residual and neglecting the interactions inside the resolvents; and no practical computation of $(1+\Omega)^{-1}$ for continuum labels. Later work — Wu's 1956 equivalence proof, the formal rearrangement-collision theory of Ekstein, Lippmann, Gerjuoy, Sunakawa, Sasakawa, and Hahn, Löwdin's nonorthogonality machinery, coupled-reaction-channel and pseudostate implementations, and the 1996 exactly solvable rearrangement model of Varma and Sudarshan — has largely absorbed the practical numerical-inversion item into standard practice, while the independence criterion, a quantitative bound on the basis-restriction trade-off, and rigorous error bounds for the first-order amplitudes remain without a published resolution. The open problem is to determine, within a precise function-space formulation, when a retained sub-union of the two partitions' eigenbases is linearly independent, and to give quantitative error bounds for the resulting first-order transition amplitudes.

Background

A rearrangement collision is a scattering process in which the grouping of the constituent particles changes between the initial and final states — for example charge transfer between two atoms, $A + B^{+} \to A^{+} + B$, where the transferred electron is bound to $A$ before the collision and to $B$ afterwards. The difficulty such processes pose for perturbation theory is structural: the natural unperturbed Hamiltonian is different before and after the collision. Writing the total Hamiltonian in two partitions,

$$H = H(A) + H(B^{+}) + V_{1} = H(A^{+}) + H(B) + V_{2},$$

where $H(A)$ is the internal Hamiltonian of atom $A$ and $V_{1}$ the interaction between $A$ and $B^{+}$ (and similarly for the second partition), the initial channel is an eigenstate of $H(A)+H(B^{+})$ while the final channels are eigenstates of $H(A^{+})+H(B)$. The corresponding eigenfunction sets $\{\phi_{\lambda}\}$ and $\{\chi_{\mu}\}$ (with discrete and continuous labels $\lambda,\mu$; continuum eigenfunctions are delta-function normalized, so that the inner product of two continuum eigenfunctions is a Dirac delta function of the label difference) are each orthonormal within themselves but mutually non-orthogonal, since they diagonalize different operators on the same configuration space.

In 1955, P. A. M. Dirac proposed a first-order scheme for exactly this situation. Given an initial eigenstate $\psi_{0}$ with energy $E$, the correction $\psi_{1}$ is obtained by expanding the source term $-V_{1}\psi_{0}$ in a retained sub-basis drawn from the union of the two sets,

$$-V_{1}\psi_{0} = \sum\int a_{\lambda}\,\phi_{\lambda}\,d\lambda + \sum\int b_{\mu}\,\chi_{\mu}\,d\mu + \psi_{\mathrm{res}},$$

where the sums run over the retained labels and the residual $\psi_{\mathrm{res}}$ is the component of the source term orthogonal to every retained basis function; the component orthogonal to the full union of the two complete families vanishes, since that union spans the configuration space. Projecting onto each basis function yields coupled linear integral equations

$$a_{\lambda} + \int \omega_{\lambda\mu}\, b_{\mu}\, d\mu = -\int \bar{\phi}_{\lambda}\, V_{1}\psi_{0}\, dq, \qquad b_{\mu} + \int \omega_{\mu\lambda}\, a_{\lambda}\, d\lambda = -\int \bar{\chi}_{\mu}\, V_{1}\psi_{0}\, dq,$$

with overlap kernels $\omega_{\lambda\mu} = \int \bar{\phi}_{\lambda}\chi_{\mu}\,dq$ and $\omega_{\mu\lambda} = \overline{\omega_{\lambda\mu}}$. In block-matrix form $(1+\Omega)Y = -Z$, where the vector $Y$ collects the coefficients $a_{\lambda}, b_{\mu}$ and $Z$ the source integrals. Dirac showed, in the paper's heuristic continuous-index formalism, that the operator $1+\Omega$ is invertible if and only if the union $\{\phi_{\lambda}\}\cup\{\chi_{\mu}\}$ is linearly independent. A resolvent is the operator inverse $(H-E)^{-1}$ (a Green's function); the scheme approximates its action on each basis function by $(E_{1\lambda}-E)^{-1}\phi_{\lambda}$ and $(E_{2\mu}-E)^{-1}\chi_{\mu}$, where $E_{1\lambda}$ and $E_{2\mu}$ are the unperturbed eigenvalues of $\phi_{\lambda}$ and $\chi_{\mu}$ under their respective partition Hamiltonians, i.e. it neglects $V_{1},V_{2}$ when inverting $H-E$ on those functions, and selects outgoing scattered waves by the standard scattering-theory replacement of each resolvent factor with $(E_{1\lambda}-E)^{-1} - i\pi\delta(E_{1\lambda}-E)$ (and similarly for $\mu$). The transition probabilities are read off from $|a_{\lambda}|^{2}$ and $|b_{\mu}|^{2}$ with suitable normalization factors.

The scheme was left unvalidated in four respects: (1) no proof that the union of the two sets is linearly independent (equivalently, that $1+\Omega$ is invertible) in concrete collision problems of practical interest; (2) no prescription for selecting an independent sub-basis when independence fails — the suggested remedy of discarding functions, for instance those corresponding to highly excited atomic states, is unquantified, and removing functions enlarges the residual $\psi_{\mathrm{res}}$; (3) no error bounds for the two approximations — discarding $\psi_{\mathrm{res}}$ and neglecting $V_{1},V_{2}$ in the resolvents — so the convergence of the first-order amplitudes is uncontrolled; and (4) no practical computation of $(1+\Omega)^{-1}$ or numerical implementation of the integral equations for continuum labels.

The surrounding context: the Born approximation (first order in the interaction) and the Lippmann–Schwinger equation (the standard integral equation satisfied by scattering states) assume a single unperturbed Hamiltonian, so rearrangement collisions motivated the multichannel formal scattering theory of the 1950s (Ekstein; Lippmann; Gerjuoy's review; Sunakawa; Sasakawa's projection-operator theory). In that literature, channel functions belonging to different partitions are treated as a non-orthogonal channel basis, and coupled equations carry the resulting overlap (metric) matrix. Ta-You Wu showed already in 1956 that Dirac's scheme, correctly formulated, reproduces exactly the usual first-order rearrangement amplitudes, so the accuracy question of the scheme is the classical, still-uncontrolled accuracy question for such amplitudes (the distorted-wave Born approximation, DWBA, is the standard first-order scheme in which the amplitude is computed to first order in the difference between the full and a chosen distorting interaction). For finite non-orthogonal bases, Löwdin's canonical orthogonalization — diagonalizing the overlap matrix and discarding eigenvectors with eigenvalues below a threshold — is the standard practical cure for near-linear dependence. Modern computational practice — coupled-reaction-channel equations in nuclear transfer (coupled-channel equations whose expansion includes rearrangement channels), two-center pseudostate close-coupling (a pseudostate is a discrete square-integrable basis function representing a continuum channel; close-coupling expands the scattering wave function in channel eigenstates and truncates) and R-matrix propagation (matching inner-region discrete expansions to outer-region scattering solutions) in atomic collisions — inverts overlap matrices routinely, but validates convergence empirically rather than by bounds. Varma and Sudarshan (1996), in an exactly solvable field-theoretic rearrangement model, argued that the non-orthogonality of channel bases is an artifact of the channel-Hamiltonian splitting itself, which they deemed fundamentally flawed, and constructed orthogonal physical channel states; their critique reframes rather than resolves the question within Dirac's framework.

Problem Statement

Determine the validity and accuracy of Dirac's 1955 mixed-basis first-order perturbation scheme for rearrangement collisions. In the two-partition setting $H = H(A)+H(B^{+})+V_{1} = H(A^{+})+H(B)+V_{2}$, with eigenfunction sets $\{\phi_{\lambda}\}$ of $H(A)+H(B^{+})$ and $\{\chi_{\mu}\}$ of $H(A^{+})+H(B)$ and overlap kernels $\omega_{\lambda\mu} = \int\bar{\phi}_{\lambda}\chi_{\mu}\,dq$:

(1) Linear independence and invertibility. For the class of two-center Coulomb rearrangement problems, charge transfer in atom–atom collisions being the paradigmatic case, in a precise function-space formulation of the physical problem (specifying the coefficient spaces for the continuous labels $\lambda,\mu$ and the operator setting for $\Omega$; independence is understood through the overlap operator, a (sub-)union being independent precisely when its restricted $1+\Omega$ is invertible), and for a proper sub-union of selected functions drawn from the two complete eigenfamilies (a truncated sub-basis, as in any finite or discretized working basis), resolve the following single determination, stated below in two equivalent formulations: (i) Give a criterion for independence of the sub-union in the continuum function-space formulation of the two-center Coulomb problem (finite or discretized truncations being only its computational instance) — an explicit spectral condition on the restricted overlap operator $\Omega$, stated in terms of the label spaces and checkable from the overlap kernels and unperturbed spectral data without inverting the restricted operator, that is necessary and sufficient for independence, going strictly beyond the bare invertibility of the restricted $1+\Omega$ or the nonsingularity of the retained finite matrix and beyond any merely sufficient small-overlap bound — and use it to decide independence or dependence for the retained sub-unions of at least one concrete two-center Coulomb configuration. (ii) Classify the dependent proper sub-unions: give a closed-form description of which selections from the two families are dependent (equivalently, of the kernel of the restricted $1+\Omega$), and exhibit the dependent sub-unions explicitly — a classification with stated structure.

(2) Controlled basis restriction. When independence fails, give a general prescription for which functions to discard from the union — one that, within a prescribed finite or numerically conditioned truncation that keeps both partitions represented, retains a maximal independent sub-basis (so the retained set is rich in each partition rather than a minimal selection) — together with a quantitative bound, stated in terms of the discarded functions, on the resulting growth of the residual $\psi_{\mathrm{res}}$ (the component of the source term orthogonal to every retained basis function); this restriction-level bound is distinct from, and feeds into, the amplitude error required in (3).

(3) Error bounds and convergence. Derive quantitative bounds — in terms of norms of the interactions $V_{1},V_{2}$, the residual $\psi_{\mathrm{res}}$, and spectral data of the two unperturbed Hamiltonians — on the error of the first-order transition amplitudes obtained by discarding $\psi_{\mathrm{res}}$ and by approximating $(H-E)^{-1}\phi_{\lambda} \approx (E_{1\lambda}-E)^{-1}\phi_{\lambda}$ and $(H-E)^{-1}\chi_{\mu} \approx (E_{2\mu}-E)^{-1}\chi_{\mu}$ (with $E_{1\lambda}$, $E_{2\mu}$ the unperturbed eigenvalues of $\phi_{\lambda}$, $\chi_{\mu}$), together with conditions under which the first-order scheme converges.

The common objective is single: to turn the mixed-basis first-order scheme into a validated approximation with controlled accuracy for rearrangement-collision transition probabilities. Deliverables (1)–(3) jointly serve this objective: (1) fixes when the scheme is well-posed, (2) restores well-posedness when it is not, and (3) certifies the accuracy of the resulting amplitudes. The practical computation of $(1+\Omega)^{-1}$ and the numerical implementation of the coupled integral equations — the fourth gap left by the source — has largely been discharged by modern coupled-channel and pseudostate technology. Because Wu (1956) showed the scheme coincides with the usual first-order rearrangement treatment when correctly formulated, a solution to (3) is simultaneously an error theory for standard first-order (Born/DWBA-type) rearrangement amplitudes.

References

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