Quasi Exactly Solvable Difference Equations

Several explicit examples of quasi exactly solvable `discrete’ quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/sin^2x potential deformed by a cos2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.

💡 Research Summary

The paper investigates quasi‑exactly solvable (QES) systems within the framework of discrete quantum mechanics (DQM), where the Hamiltonian is built from shift operators rather than differential operators. Starting from two well‑known exactly solvable one‑dimensional DQM models – the harmonic oscillator (with an optional centrifugal term) and the Pöschl‑Teller potential (V(x)=g(g-1)/\sin^{2}x) – the authors introduce specific deformations that preserve a finite‑dimensional invariant subspace of the Hilbert space.

For the harmonic oscillator, a sextic term (\lambda x^{6}) is added to the potential, yielding a deformed potential (V_{\text{HO}}^{\text{def}}(x)=\frac{1}{2}x^{2}+\lambda x^{6}+g(g-1)/x^{2}). In the Pöschl‑Teller case, a cosine perturbation (h\cos 2x) is introduced, giving (V_{\text{PT}}^{\text{def}}(x)=g(g-1)/\sin^{2}x+h\cos 2x). The corresponding shift operators (\mathcal{A}) and (\mathcal{A}^{\dagger}) are modified accordingly, but retain the generic form (\mathcal{A}=e^{\pm\partial/2}\sqrt{V^{\text{def}}(x)}-e^{\mp\partial/2}\sqrt{V^{\text{def}}(x)}).

A key technical device is the introduction of a sinusoidal coordinate (\eta(x)) – (\eta=x^{2}) for the oscillator and (\eta=\cos 2x) for the Pöschl‑Teller model. Expressed in terms of (\eta), the deformed Hamiltonian becomes a second‑order difference operator, \