Classify non-terminating solutions of the multi-dimensional harmonic Darboux chain

Classify all non-terminating homogeneous polynomial solution families of the multi-dimensional harmonic Darboux chain τ_n Δ τ_{n+1} − 2(∇τ_{n+1} · ∇τ_n) + τ_{n+1} Δ τ_n = 0 in arbitrary spatial dimension, thereby identifying all τ-functions that yield algebraically integrable Schrödinger operators beyond the known one- and two-dimensional families.

Background

The paper discusses a higher-dimensional generalization of the Darboux chain to harmonic chains of homogeneous polynomials, which generate potentials for algebraically integrable Schrödinger operators. In one dimension, the Adler–Moser polynomials provide the complete set of non-terminating solutions; in two dimensions, soliton-related τ-functions exhaust all non-terminating solutions in the class of homogeneous polynomials.

Beyond these cases, the authors note that classifying all non-terminating solutions in arbitrary dimensions is an open problem. This relates to understanding the full structure of algebraically integrable Schrödinger operators via polynomial τ-functions across dimensions.