Solve the quantum Gérard–Grellier Hamiltonian by explicitly constructing all eigenfunctions and eigenvalues

Construct an explicit solution of the quantum Gérard–Grellier Hamiltonian H=\tfrac12\sum_{n+m=k+l} a_n^† a_m^† a_k a_l by providing a complete, closed-form description of all its eigenfunctions and corresponding eigenvalues across all (N,M) blocks.

Background

The authors develop energy bounds, families of eigenvectors, ladder operators, and partial diagonalizations revealing striking square-integer patterns, but a comprehensive analytical solution is not yet achieved.

A full solution—an explicit construction of all eigenfunctions and spectra—would resolve the central outstanding challenge and firmly establish the superintegrability of the model in a complete, constructive manner.