Closed-form solvability of the inverse Feshbach reconstruction for arbitrary finite dimension increase

Establish that, for every positive integer K, all matrix elements of a partially tridiagonalized full-space Hamiltonian H^(M+K) can be expressed as ratios of polynomial functions of the effective-Hamiltonian input data E_alpha and G_alpha, thereby proving closed-form solvability of the inverse Feshbach reconstruction beyond the explicitly verified cases.

Background

The paper studies the inverse Feshbach problem of reconstructing a finite-dimensional full-space Hamiltonian H^M+K from the energy-dependent effective Hamiltonian H_eff^M(E). Under an enhanced tridiagonality assumption, the unknown diagonal entries a_j and off-diagonal products rho_j are determined implicitly by 2K+1 coupled polynomial equations involving sampled values of the effective-Hamiltonian function G(E).

For K=1, K=2, and K=3, the paper obtains exact rational-function solutions, although the expressions become increasingly complicated. The authors conjecture that the same rational-function solvability persists for every finite K; this remains unresolved in general because only finitely many cases are explicitly constructed.