Inverse Feshbach's problem: Solvability and solutions

AI summary

Overview

Research area: Quantum theory of effective Hamiltonians — specifically the mathematical structure of the Feshbach/Löwdin model-space reduction and its inverse. The paper sits at the boundary between quantum many-body methodology and constructive (symbolic) algebra.

Technical level: Advanced. The paper is written for readers comfortable with projector formalism, continued fractions, Jacobi matrices, and computer-algebraic elimination (Gröbner bases).

Scope (one sentence): The paper argues, against the author's own earlier claim, that a partially tridiagonalized full-space Hamiltonian can be reconstructed in closed form from the single energy-dependent matrix element of its effective (Feshbach) partner, and demonstrates this constructively up to K = 3 while proposing a recursive scheme for general K.

What This Paper Is About

In quantum mechanics one often replaces a large Hamiltonian H by a smaller, energy-dependent "effective" Hamiltonian H_eff(E) that reproduces the same spectrum inside a chosen model space. Going forward (H → H_eff(E)) is routine; going backward (H_eff(E) → H) has been regarded as practically unfeasible. The paper's goal is to show that this inverse reconstruction is in fact solvable in closed algebraic form, at least under a strong simplifying assumption about the structure of H.

Key Contributions

1. Reversal of the author's prior conclusion. The earlier skepticism, published in Phys. Lett. A 556 (2025) 130816, held that reconstruction stays prohibitively difficult even after maximal simplification of the target matrix H (assumed partially tridiagonalized). The present paper explicitly states that this skepticism "is disproved."

2. A closed-form conjecture (Conjecture 3). At any positive integer K, every to-be-reconstructed matrix element of the full-space Hamiltonian is conjectured to be expressible as a ratio of two polynomial functions of the dynamical input parameters E_α and G_α (equivalently of the auxiliary variables x_m and y_n).

3. Constructive exact solutions at K = 1, 2 and 3. K = 1 is solved by Lemma 1 with fully explicit formulae; K = 2 is solved by Gröbner elimination giving closed-form expressions in Eqs. (37)–(40); K = 3 passes the same test but produces formulae of impractical length.

4. A recursive reformulation for general K. Instead of fully explicit element-by-element definitions, the paper proposes a recursive evaluation scheme whose specifying formulae acquire a "stabilized" canonical form for K ≥ K_stab, where the threshold K_stab = K_stab(j,k) depends on the matrix-element subscripts j and k.

Main Findings

• The reduction is localized to one matrix element. With the "enhanced tridiagonality" assumption — i.e., Q^(+) H Q^(+) tridiagonal, where Q^(+) = |M⟩⟨M| + Q = Σ_{m=M}^{M+K} |m⟩⟨m| — the effective Hamiltonian differs from the plain truncated matrix H^(M) only in the single entry [H_eff(E)]_MM = 𝒢(E) + E, with 𝒢(E) = 1/f_0(E) and f_0 = [I/(Q^(+) H Q^(+) − E)]_MM.

• The inverse problem becomes 2K + 1 polynomial equations. Sampling the identity at 2K + 1 energies E_α (α = 0, 1, …, 2K) converts the reconstruction into a coupled nonlinear algebraic system, Eq. (18): 𝒢(E_α) = 1/f_0(E_α), or equivalently the determinant condition of Lemma 2.

• A normalization is required to make the solution unique. In the raw ansatz the off-diagonal entries enter only through the products ρ_n = b_n c_{n+1}. The paper therefore fixes the lower diagonal to ones and treats the c_n as a K-plet of free parameters, yielding the isospectral partner form of Eq. (17).

• K = 1 is elementary. Three equations (α = 0, 1, 2) determine a_0, a_1 and ρ_0; the solution is a_1 = y_1, a_0 = −x_1 − y_1, ρ_0 = −x_1 y_1 − x_2 − y_1².

• K = 2 is compact. Five equations reduce to a linear five-by-five system (Eq. 33) relating ten empirical inputs {E_0, …, E_4, G_0, …, G_4} to five compact inputs {x_1, x_2, x_3, y_1, y_2}; Gröbner elimination then returns closed-form expressions for a_0, ρ_0, a_1, ρ_1 and a_2.

• K = 3 is solvable but unwieldy. Seven coupled polynomial equations are handled by the same Gröbner-basis algorithm, and Conjecture 3 survives the test — but only three formulae (for a_0, ρ_0, a_1) are short, while the expression for ρ_1 "did not fit in one standard page," with a_2 similarly long. This length explosion, not insolubility, is identified as the real obstacle at K = 3.

• No large-energy asymptotics needed. Unlike conventional spectral-analytic treatments of the inverse problem, the method does not require knowledge of the E → ∞ asymptotics of the m-function 𝒢(E); all input is the function's values on an interval (E_min, E_max).

• The stabilization bound is not sharp. The paper notes that K_stab(j,k) "appeared not to be always sufficiently sharp," which opens the questions addressed in Sections 8, 9 and 10.

Methodology in Plain English

The author starts from a Hamiltonian written in a basis chosen so that the part lying outside the model space is tridiagonal, and so that the coupling between the model space and the outside world passes through a single "doorway" state. This makes the effective Hamiltonian almost identical to the naive truncated matrix: only one entry, at position (M, M), carries all the energy dependence.

The reconstruction then rests on a simple observation. That one entry is a rational function of the unknown tridiagonal block — a continued fraction built from the unknowns a_j and ρ_k. If you evaluate this function at enough different energies, you get as many equations as unknowns: 2K + 1 equations for the K + 1 values of a_j and the K values of ρ_k. These are polynomial equations, not linear ones, so the author rewrites them in terms of aggregated variables (x_m, y_n) where the system becomes linear and can be inverted with ordinary matrix algebra. The remaining nonlinear step — recovering the individual a_j and ρ_k from the aggregated variables — is done with Gröbner-basis elimination, a computer-algebra technique for solving polynomial systems exactly.

At K = 1 and K = 2 this closing step yields tidy rational expressions. At K = 3 it still succeeds in principle but the expressions become page-long. That motivates the shift in later sections from "explicit formula for each element" to "recursive rule for generating each element," which the author reports takes a stable, canonical shape beyond an element-dependent threshold K_stab.

Why This Matters

The paper addresses a foundational consistency question: energy-dependent effective Hamiltonians are often fitted to data with no guarantee that any underlying energy-independent quantum Hamiltonian exists. A constructive inverse map turns that guarantee into something checkable.

• Nuclear theory: Feshbach-inspired effective interactions are a standard tool in nuclear structure calculations; the paper cites Refs. [3, 4] in this context.
• Quantum chemistry: Löwdin-style isospectral reduction of large configuration-space matrices is used routinely (Refs. [5, 6]).
• PT-symmetric and non-Hermitian models: effective-Hamiltonian constructions appear in PT-symmetric quantum models [7] and non-commutative quantum mechanics [8].
• Relativistic quantum physics and information theory: the paper lists [9] and [10] as further application areas of the same replacement.

Industry relevance: the paper itself reports none — no datasets, no benchmarks, no numerical experiments, no software performance figures. Its relevance to industrial modeling (reactor physics, electronic-structure codes, quantum-device design) is indirect, resting on the broader use of model-space reduction in those fields; the paper makes no such claims and states no numbers.

Future Directions

• Sharpening the stabilization threshold. The bound K_stab(j,k), above which the recursive formulae take canonical form, was found not always sharp; Sections 8–10 are described as offering only a partial answer, leaving the precise dependence on j and k open.
• Proving or refuting Conjecture 3. The closed-form rationality claim is verified constructively only at K = 1, 2 and 3; a general proof for arbitrary K is not given.
• Taming the expression growth at K ≥ 3. Finding a representation that avoids the page-long formulae — presumably through the recursive scheme rather than explicit elimination — is the natural next technical step.
• Testing beyond the enhanced-tridiagonality assumption. The entire construction leans on partial tridiagonalization of H and on a single doorway coupling; how far the approach survives weaker structural assumptions is not addressed in the material available.

Target Audience

Mathematical and theoretical physicists working on effective Hamiltonians, model-space reduction, and inverse spectral problems; nuclear-structure and quantum-chemistry methodologists who build energy-dependent effective interactions; and applied mathematicians interested in constructive solutions of coupled polynomial systems arising from continued fractions. Readers seeking numerical benchmarks or experimental validation will not find them here.

Authors’ abstract

Given a certain specific, $M$ by $M$ matrix form of the Feshbach's effective (i.e., energy-dependent) Hamiltonian $H^{(M)}_{e\!f\!f}(E)$, the inverse-problem reconstruction of the full-space, $N$ by $N$ matrix Hamiltonian $H^{(N)}$ 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 $H^{(N)}$ is found feasible at not too large $K = N-M$.