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 η_enh propto \(z - z_c\)^-1/2, achieving 35-fold enhancement at δ= 0.001. Comprehensive numerical verification confirms: (i) all seven bound states for strength parameter R = 10 satisfying the constraint u² + v² = R²; (ii) exact agreement between theoretical band gap formula E_g = 2πhbar v_F/(3L) and empirical relation E_g = 1.38/L eVcdotnm; (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.