Lambert W Function Framework for Graphene Nanoribbon Quantum Sensing: Theory, Verification, and Multi-Modal Applications
- URL: http://arxiv.org/abs/2601.10767v1
- Date: Thu, 15 Jan 2026 08:32:47 GMT
- Title: Lambert W Function Framework for Graphene Nanoribbon Quantum Sensing: Theory, Verification, and Multi-Modal Applications
- Authors: F. A. Chishtie, K. Roberts, N. Jisrawi, S. R. Valluri, A. Soni, P. C. Deshmukh,
- Abstract summary: We establish a rigorous mathematical framework connecting graphene nanoribbon quantum sensing to the Lambert W function through the finite square well (FSW) analogy.<n>We demonstrate that operating near the branch point at $z = -1/e$ yields sensitivity enhancement factors scaling as $_textenh propto (z - z_c)-1/2$, achieving 35-fold enhancement at $= 0.001$.
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- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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 $η_{\text{enh}} \propto (z - z_c)^{-1/2}$, 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 $u^2 + v^2 = R^2$; (ii) exact agreement between theoretical band gap formula $E_g = 2π\hbar v_F/(3L)$ 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.
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