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. Operating near the branch point singularity at $z=-1/e$ yields sensitivity enhancement factors scaling as $(z-z_c)^{-1/2}$, achieving 35-fold enhancement when the operating point lies within $\delta=0.001$ of the branch point. Comprehensive numerical verification confirms: (i) all seven bound states for strength parameter $R=10$ satisfy the constraint $u^2+v^2=R^2$ to machine precision; (ii) the theoretical band gap formula $E_g=2\pi\hbar v_F/(3W)$ is analytically equivalent to the independently determined empirical relation $E_g=1.38/W$~eV$\cdot$nm, establishing the validity of the FSW-GNR analogy; (iii) a universal sensitivity factorization $S_X = \mathcal{G}_k \cdot \eta_{\rm enh} \cdot \mathcal{P}_X$ applies 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 analytically predictable design principles for next-generation graphene quantum sensors. The framework is analytic and predictive rather than microscopic or experimental: band-structure and adsorption parameters are taken as inputs from tight-binding, first-principles, and experimental sources, and the framework returns closed-form sensitivity and design relations built upon them. Reported detection limits are labelled throughout as either literature-demonstrated device values or values predicted by the present framework.