Inverse Feshbach's problem: Solvability and solutions

Abstract: Given a certain specific, $M$ by $M$ matrix form of the Feshbach's effective (i.e., energy-dependent) Hamiltonian $H<sup>{(M)}_{e!f!f}(E)$, the inverse-problem reconstruction of the full-space, $N$ by $N$ matrix Hamiltonian $H<sup>{(N)}$ is considered and reduced to the solution of a coupled set of polynomial algebraic equations. Using computer-assisted symbolic manipulations, an explicit algebraic reconstruction of $H<sup>{(N)}$ is found feasible at not too large $K = N-M$.

• The paper presents a constructive and detailed methodology to solve the inverse Feshbach problem for reconstructing a full-space Hamiltonian from a known energy-dependent effective Hamiltonian.
• Key to the solution is a basis choice that renders the Hamiltonian enhanced-tridiagonal, simplifying the reconstruction process via a Weyl m-function that confines energy dependence into key matrix elements.
• Implementing compact recursive formulas ensures solvability at larger K without direct Gröbner expressions, providing practical solutions to previously considered unsolvable scenarios.

Setting: effective Hamiltonians and the reconstruction question

The Feshbach projection formalism replaces a full-space Hamiltonian $H$ acting in an $N$-dimensional Hilbert space by its model-space projection, and compensates for the truncation through an energy-dependent effective Hamiltonian $H_{\mathrm{eff}}^{(M)}(E)$ acting on an $M$-dimensional subspace. The forward direction $H \to H_{\mathrm{eff}}(E)$ is routine and underlies applications from nuclear theory and quantum chemistry to PT-symmetric models and energy-dependent potentials. The inverse problem — reconstructing $H^{(M+K)}$ from a known $H_{\mathrm{eff}}^{(M)}(E)$ — is motivated by phenomenology in which ad hoc energy-dependent matrices are fitted to data without any guarantee that a compatible full-space quantum Hamiltonian exists (2608.18600). A companion letter by the same author had implemented this reconstruction only at the first nontrivial case $K = N - M = 1$, and expressed skepticism that larger $K$ could be handled in closed form. The present paper disproves that skepticism constructively.

Structure of the problem

The key simplification is a basis choice rendering the Hamiltonian "enhanced-tridiagonal": the block $QHQ$ and even $N$0, where $N$1, are tridiagonal. This corresponds to a doorway-state-mediated decoupling of subspaces. Under this ansatz the entire energy dependence of the effective Hamiltonian is confined to a single matrix element,

$N$2

where $N$3 and $N$4 admits a continued-fraction representation — mathematically the Weyl $N$5-function of a semi-infinite Jacobi matrix.

A crucial non-uniqueness must be resolved first: the off-diagonal entries $N$6 and $N$7 enter $N$8 only via products $N$9. The reconstruction is therefore made unique by fixing the lower diagonal to unity and working with an isospectral partner matrix containing $H_{\mathrm{eff}}^{(M)}(E)$0 and $H_{\mathrm{eff}}^{(M)}(E)$1 as unknowns — altogether $H_{\mathrm{eff}}^{(M)}(E)$2 quantities. Assuming knowledge of $H_{\mathrm{eff}}^{(M)}(E)$3 at $H_{\mathrm{eff}}^{(M)}(E)$4 sample energies $H_{\mathrm{eff}}^{(M)}(E)$5, one obtains the coupled polynomial system

$H_{\mathrm{eff}}^{(M)}(E)$6

Notably, the method deliberately avoids the standard spectral-theoretic route requiring large-energy asymptotics of the $H_{\mathrm{eff}}^{(M)}(E)$7-function.

Explicit solutions and the failure of brute force

At $H_{\mathrm{eff}}^{(M)}(E)$8 the system is linearized into three linear equations in auxiliary variables $H_{\mathrm{eff}}^{(M)}(E)$9 (elementary symmetric functions of the unknowns), solvable by matrix inversion followed by trivial back-substitution. At $M$0 the same strategy works: a five-by-five linear inversion converts the ten empirical inputs $M$1 into $M$2, and Gröbner-basis elimination yields closed rational-function formulae for all five unknowns. This prompted the paper's central conjecture: at any positive integer $M$3, all reconstructed matrix elements are ratios of polynomials in the dynamical input parameters.

Direct verification at $M$4 confirmed the conjecture but exposed the practical obstacle. The explicit formulae become prohibitively long — the expression for $M$5 does not fit on a printed page, and $M$6 requires roughly 25 lines — so the raw Gröbner output is essentially computer-memory-bound rather than usable. This is precisely where the earlier skepticism had arisen, and it is here that the paper changes tack.

Compact recursive reformulation

The remedy rests on expanding the characteristic determinants of the tridiagonal blocks in powers of their spectral parameters, generating coefficient families $M$7 (from $M$8), $M$9 (from $H \to H_{\mathrm{eff}}(E)$0), and $H \to H_{\mathrm{eff}}(E)$1, subject to sign conventions $H \to H_{\mathrm{eff}}(E)$2, $H \to H_{\mathrm{eff}}(E)$3. Sums such as $H \to H_{\mathrm{eff}}(E)$4, $H \to H_{\mathrm{eff}}(E)$5, $H \to H_{\mathrm{eff}}(E)$6 eliminate higher-index unknowns and yield compact recurrences running forward in the index. The resulting formulae stabilize: beyond an element-dependent threshold $H \to H_{\mathrm{eff}}(E)$7 they become $H \to H_{\mathrm{eff}}(E)$8-independent up to oscillating signs. Representative results:

Quantity	Stabilized form
$H \to H_{\mathrm{eff}}(E)$9	$H^{(M+K)}$0
$H^{(M+K)}$1	$H^{(M+K)}$2
$H^{(M+K)}$3	$H^{(M+K)}$4
$H^{(M+K)}$5	$H^{(M+K)}$6

Two further conjectures organize these observations. First, every matrix element $H^{(M+K)}$7 admits a comparatively compact rational recurrence in terms of its predecessors. Second — described as one of the most striking results — the number of terms in each forward recurrence ceases growing at $H^{(M+K)}$8 for diagonal elements $H^{(M+K)}$9 and $H_{\mathrm{eff}}^{(M)}(E)$0 for off-diagonal products $H_{\mathrm{eff}}^{(M)}(E)$1. If correct, this guarantees user-friendliness at arbitrarily large $H_{\mathrm{eff}}^{(M)}(E)$2. It should be noted that these stabilizations are supported by constructive calculations at small $H_{\mathrm{eff}}^{(M)}(E)$3 plus backward-insertion checks, not yet by rigorous induction proofs, which the author indicates would be feasible using elementary determinant properties.

Optimal strategy at fixed dimension

For a fixed target $H_{\mathrm{eff}}^{(M)}(E)$4, the stabilization bound $H_{\mathrm{eff}}^{(M)}(E)$5 exceeds the largest needed index, so forward recurrences alone are wasteful near $H_{\mathrm{eff}}^{(M)}(E)$6. The paper proposes matching forward-running and backward-running recurrences mid-range. At $H_{\mathrm{eff}}^{(M)}(E)$7 this reduces the final self-consistency condition to a single linear equation for $H_{\mathrm{eff}}^{(M)}(E)$8. At $H_{\mathrm{eff}}^{(M)}(E)$9, treating the last three equations as backward recurrences after the forward-determined $K = N - M = 1$0 have been inserted again reproduces the tabulated $K = N - M = 1$1. An alternative determinantal scheme introduces auxiliary variables $K = N - M = 1$2 from the third expansion family; its systematic development at $K = N - M = 1$3 is explicitly deferred as future work.

Limitations and open questions

Several qualifications bear directly on the results. The conjecture of universal closed-form rational solvability has been verified only up to $K = N - M = 1$4 by exhaustive computation, with partial structural evidence beyond; no general proof exists. Similarly, the stabilization thresholds $K = N - M = 1$5 are empirically extrapolated and admitted to be possibly non-sharp. The entire construction presupposes the enhanced-tridiagonality (doorway-state) hypothesis on the target Hamiltonian and exact knowledge of $K = N - M = 1$6 on an empirical energy interval; robustness to noisy or incomplete sampling of $K = N - M = 1$7 is not addressed. Uniqueness holds only modulo the fixed normalization of the lower diagonal, which absorbs the gauge freedom in $K = N - M = 1$8. Finally, the systematic theory of matched forward/backward constructions at general $K = N - M = 1$9, and the rigorous induction proofs of the stabilized recurrences, remain open.

Conclusion

The paper reverses the previously stated pessimism about the inverse Feshbach problem. Under a partially tridiagonal ansatz, reconstruction of the full-space Hamiltonian reduces to $K$0 coupled polynomial equations whose solutions exist in closed rational form; although the direct Gröbner-based expressions are unusable already at $K$1, rewriting them as forward recurrences produces compact, apparently stabilizing definitions valid at all finite $K$2. Together with a matched-recurrence strategy optimized for fixed dimensions, this renders the reconstruction $K$3 practically tractable, thereby providing an existence-and-consistency check for phenomenological models built on ad hoc energy-dependent effective Hamiltonians.