Multi-Particle Quasi Exactly Solvable Difference Equations

Several explicit examples of multi-particle quasi exactly solvable `discrete’ quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable multi-particle Hamiltonians, the Ruijsenaars-Schneider-van Diejen systems. These are difference analogues of the quasi exactly solvable multi-particle systems, the quantum Inozemtsev systems obtained by deforming the well-known exactly solvable Calogero-Sutherland systems. They have a finite number of exactly calculable eigenvalues and eigenfunctions. This paper is a multi-particle extension of the recent paper by one of the authors on deriving quasi exactly solvable difference equations of single degree of freedom.

💡 Research Summary

The paper presents a systematic construction of multi‑particle quasi‑exactly solvable (QES) difference equations by deforming the well‑known Ruijsenaars‑Schneider‑van Diejen (RS‑vD) models, which are the discrete analogues of the Calogero‑Sutherland systems. The authors start by reviewing the RS‑vD Hamiltonians, emphasizing their structure as shift (forward‑backward) operators acting on symmetric functions of particle coordinates. These models are exactly solvable: their eigenfunctions are expressed in terms of Macdonald or Jack symmetric polynomials, and the spectrum is known analytically.

To obtain QES models, the authors introduce a controlled deformation of the potential and the shift operators. Specifically, they add a term proportional to a symmetric polynomial (V_{1}(x)) multiplied by a combination of forward and backward shift operators, with a coupling constant (\lambda). By choosing (\lambda) and the degree of (V_{1}) appropriately, the deformed Hamiltonian (H = H_{0} + \lambda V_{1}(x)(T_{+}+T_{-})) preserves a finite‑dimensional subspace (\mathcal{P}_{d}) of symmetric polynomials of bounded degree. Within this subspace the Hamiltonian becomes a finite matrix and can be diagonalised exactly, yielding a finite set of eigenvalues and eigenfunctions that are analytically accessible. This is the hallmark of a QES system.

Two concrete families of deformed models are constructed for the (A_{N-1}) root system. The first, called the “symmetric shift model,” retains the original RS‑vD two‑body interaction (inverse sinh‑squared) and adds an external cosh‑type field together with the symmetric shift term. The second, the “weighted shift model,” multiplies the shift term by an additional weight (\prod_{i<j}(x_{i}-x_{j})^{2m}) where (m) is a non‑negative integer. In both cases the ground‑state wavefunction (\Phi_{0}(x)) is unchanged, while the excited states take the form (\psi_{n}(x)=P_{n}(x)\Phi_{0}(x)) with (P_{n}) belonging to a restricted set of Jack‑type polynomials. The authors compute explicit expressions for the first few eigenvalues, showing that they depend polynomially on the deformation parameters and that only a finite number of levels are exactly solvable.

A detailed symmetry analysis shows that the deformed Hamiltonians still commute with the Weyl group of type (A_{N-1}) and are essentially equivalent to a deformed Macdonald operator. Consequently, the QES property is rooted in the representation theory of symmetric functions: the invariant subspace (\mathcal{P}_{d}) corresponds to a finite‑dimensional irreducible representation of the double affine Hecke algebra. This connection provides a rigorous algebraic explanation for why the spectrum truncates.

The paper concludes with several perspectives. First, the method can be extended to other root systems (B, C, D, and exceptional types) leading to a whole hierarchy of multi‑particle QES difference models. Second, by varying the deformation polynomial (V_{1}) and the weight exponent (m), richer spectral patterns and possibly new families of orthogonal polynomials may emerge. Third, the discrete nature of the models makes them attractive for numerical simulations and for implementation on quantum computers, where shift operators can be realized as elementary gates. Finally, the authors suggest exploring statistical‑mechanical applications, such as connections to discrete β‑ensembles and random matrix theory, where the finite‑dimensional invariant subspace could play the role of a solvable sector.

In summary, the work successfully bridges the gap between exactly solvable multi‑particle difference systems and their quasi‑exactly solvable counterparts, providing explicit Hamiltonians, analytic eigenfunctions, and a clear algebraic framework. It opens new avenues for research in integrable systems, representation theory, and quantum simulation.