This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $d\mu = w(\theta) \frac{ d\theta}{ 2\pi} + d\mu_s$ with Verblunsky coefficients $\alpha=\{ \alpha_n\}_{ n=0}^\infty$, distinct singular points $(\theta_k)_{ k=1}^{ \ell}$, and multiplicities $(m_k)_{ k=1}^{ \ell}$, we establish the equivalence between the entropy condition \[ \int_0^{ 2\pi} \prod_{ k=1}^{ \ell} [1 - \cos(\theta - \theta_k)]^{ m_k} \log w(\theta) \frac{ d\theta}{ 2\pi} > -\infty \] and the decomposition condition \[ \exists \beta^{ (1)}, \ldots, \beta^{ (\ell)} : \alpha = \sum_{ k=1}^\ell \beta^{ (k)} \,\, \text{ with} \,\, (S - e^{ -i\theta_k})^{ m_k} \beta^{ (k)} \in \ell^2, \,\, \beta^{ (k)} \in \ell^{ 2m_k + 2}. \] The proof synthesizes unitary transformations, discrete Sobolev-type inequalities, higher-order Szeg\H{ o} expansions, and a novel algebraic decomposition technique. Our resolution affirms that spectral theory is fundamentally local-global behavior emerges from the superposition of local resonances, each governed by its intrinsic scale.
Categories: math.SP