On the Spectral Analysis of Diamond-α Sturm–Liouville Problems on Uniform Time Scales

This paper investigates a Sturm–Liouville problem formulated in terms of the diamond-α derivative on a uniform time scale. The proposed setting combines the delta and nabla components within a unified spectral framework. A key feature of this framework is the explicit identification of the defect structure arising from diamond-α integration by parts, which distinguishes the resulting spectral theory from its classical continuous counterpart. Under appropriate regularity and uniqueness assumptions, a conditional linear-dependence criterion is established for solutions corresponding to the same eigenvalue, whereas eigenfunctions associated with distinct eigenvalues satisfy an orthogonality relation under a suitable compatibility condition. A generalized Green identity is derived, and the resulting remainder term is shown to define a skew-symmetric defect form. Moreover, an explicit estimate for this term is obtained, yielding Rα(u,v)=O(ν) as the graininess ν tends to zero under uniform boundedness assumptions. When the boundary form vanishes and the spectral gap remains uniformly separated from zero, this estimate further implies asymptotic orthogonality of eigenfunctions associated with distinct eigenvalues in the dense-limit regime. A generalized integral identity for the eigenfunctions is also established. The classical continuous structure is recovered when T=R, while the defect term vanishes in the dense-limit regime under the stated boundedness assumptions.