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 $f(W)=W{e}^W$, provides exact analytical solutions to transcendental equations governing quantum confinement. We demonstrate that operating near the branch point at $z=-1\mathrm{/}e$ yields sensitivity enhancement factors scaling as ${\eta }_{\text{enh}}\propto (z-z_c{)}^{-1\mathrm{/}2}$, 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 $E_g=2\pi \mathrm{\hslash }{v}_F\mathrm{/}(3L)$ and empirical relation $E_g=1.38\mathrm{/}L$ 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.