A Discrete Prüfer Transformation Approach to Sturm--Liouville Difference Equations and Eigenvalue Estimation

In this paper, we study regular second-order Sturm--Liouville difference equations using the discrete Pr\"ufer transformation. By representing solutions in amplitude and phase coordinates, we analyze an exact algebraic phase system that guarantees unique, monotonic phase tracking and preserves classical oscillation properties. Using this theoretical foundation, we develop a Pr\"ufer-based numerical shooting method to compute eigenvalues for discrete boundary value problems. To initialize the root-finding algorithm, we apply Gershgorin's theorem to the difference operator to establish mathematically guaranteed starting search intervals. Numerical experiments on classical benchmark problems demonstrate that the proposed method effectively isolates the discrete spectrum and converges to the exact continuous eigenvalues with second-order $\mathcal{O}(h^2)$ accuracy.