Lambert W Function Framework for Graphene Nanoribbon Quantum Sensing: Theory, Verification, and Multi-Modal Applications

We establish a rigorous mathematical framework connecting graphene nanoribbon quantum sensing to the Lambert W function through the finite square well (FSW) analogy. The Lambert W function, defined as the inverse of $f(W) = We^W$, provides exact analytical solutions to transcendental equations governing quantum confinement. We demonstrate that operating near the branch point at $z = -1/e$ yields sensitivity enhancement factors scaling as $\eta_{\text{enh}} \propto (z - z_c)^{-1/2}$, achieving 35-fold enhancement at $\delta = 0.001$. Comprehensive numerical verification confirms: (i) all seven bound states for strength parameter $R = 10$ satisfying the constraint $u^2 + v^2 = R^2$; (ii) exact agreement between theoretical band gap formula $E_g = 2\pi\hbar v_F/(3L)$ and empirical relation $E_g = 1.38/L$ eV$\cdot$nm; (iii) universal sensitivity scaling across biomedical (SARS-CoV-2, inflammatory markers, cancer biomarkers), environmental (CO$_2$, CH$_4$, NO$_2$, N$_2$O, H$_2$O), and physical (strain, magnetic field, temperature) sensing modalities. This unified framework provides design principles for next-generation graphene quantum sensors with analytically predictable performance.