Classification of Non-self-adjoint Singular Sturm–Liouville Difference Equations

This paper is concerned with non-self-adjoint singular Sturm–Liouville difference equations. By introducing a new spectral parameter, we rewrite the Sturm–Liouville difference equation as a formally self-adjoint Hamiltonian difference system. Applying the theory of the limit point and limit circle cases for this difference system, we classify the considered equation into cases I, II, and III. Two examples are illustrated to show the dependence of cases II and III on the corresponding half planes. Furthermore, the exact dependence of cases II and III on the corresponding half planes is obtained.