Inverse Feshbach’s problem: solvability and solutions

Abstract

The well known Feshbach-inspired isospectral replacement of a conventional quantum Hamiltonian (Formula presented) (Formula presented) by its energy-dependent but user-friendlier effective-Hamiltonian alternative (Formula presented) (Formula presented) (acting in a suitable subspace) is a more or less routine procedure. In contrast, the inverse problem of reconstruction (Formula presented) (Formula presented) is considered unfeasible. Recently, the skepticism has been further enhanced in Znojil (2025 Phys. Lett. A 556 130816) where we argued that the reconstruction remains prohibitively difficult even after a maximal simplification of the target matrix (Formula presented) (Formula presented) (assumed to be partially tridiagonalized). In the present paper the latter skepticism is disproved. It is conjectured (and, for a number of special cases, it is constructively demonstrated) that the ‘missing’ matrix elements of (Formula presented) (Formula presented) can be reconstructed and defined, in terms of the relevant matrix elements of (Formula presented) (Formula presented), in closed form. © 2026 The Author(s). Published by IOP Publishing Ltd. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.