Inverse Problem in the Deformation Method: A General Framework with Rigorous Analysis and Algorithms

The Bazeia--Losano--Malbouisson deformation method generates new scalar field potentials from a seed potential via a smooth deformation function. While the forward problem is well understood, this paper provides a rigorous and general analysis of the inverse problem: given desired kink masses and possibly internal mode frequencies, can one reconstruct the deformation function for an arbitrary seed? We prove a fundamental impossibility theorem for multiple kinks with alternating non‑degenerate vacua when the seed possesses only two minima. For the single‑kink case we establish a complete existence theorem: the attainable masses form an interval \((Minf,∞)\), where \(Minf\) is either zero or a positive number determined solely by the seed potential and the leaf length. The proof uses a variational characterisation and explicit constructions. We extend the method to shape‑mode constraints as a numerical optimisation problem and discuss multiple‑kink configurations with degenerate interior vacua or with seeds having more than two minima. Detailed, implementable algorithms, valid for any seed potential, are provided.