The spectral analysis of the difference quotient operator on model spaces

We conduct a spectral analysis of the difference quotient operator ${\text{Q}}_{\zeta }^u$, associated with a boundary point $\zeta \in \partial D$, on the model space $K_u$. We describe the operator's spectrum and provide both upper and lower estimates for its norm $\Vert {\text{Q}}_{\zeta }^u\Vert$, and furthermore discussing the sharpness of these bounds. Notably, the upper estimate offers a new characterization of the one-component property for inner functions.