The Hajji Hamiltonian: Complete Mathematical Derivation of Logarithmic Spectral Scaling and Excitonic Transitions (Part I: Spectral Properties and Experimental Predictions)

Title: The Hajji Hamiltonian: Complete Mathematical Derivation of Logarithmic Spectral Scaling and Excitonic Transitions (Part I: Spectral Properties and Experimental Predictions) Authors: Mohammed Hajji Abstract: We present a complete mathematical derivation of the spectral properties of the Hajji Hamiltonian H = -d²/dx² + α ln x + β, with parameters α = 8.011678 and β = -14.972991 (atomic units). These parameters are uniquely determined from (6,5) single-walled carbon nanotubes (SWCNTs) under a 15 T axial magnetic field, using the experimental values E₁ = -8.168 eV and E₂-E₁ = 4.476 eV. The derivation proceeds through four rigorous steps: (1) Bessel solution near x=0, proving ψ(x) ∝ √x J₁(2√αx) ∼ x; (2) logarithmic coordinate transformation x = eʸ mapping the problem to an Airy equation; (3) asymptotic analysis yielding energy eigenvalues E_n = α ln a_n + β, where a_n are zeros of Ai(-a_n)=0; (4) asymptotic scaling E_n ∼ (2α/3) ln n. From this we derive exponential density of states g(E) ∼ (3/2α) e^{3E/(2α)} and linear entropy-energy relation S(E) ∼ (3/2α)E. The predicted optical transitions are λ = 277 nm (E₂→E₁), 515 nm (E₃→E₂), 749 nm (E₄→E₃), and 983 nm (E₅→E₄), with nearly equal intensities, high polarization (I∥/I⊥ ≈ 10³), and specific magnetic field dependence. Comparison with existing experiments shows 99.6% agreement for the first two transitions. This work establishes the Hajji Hamiltonian as a new paradigm for singular quantum potentials with direct experimental relevance to carbon nanotube spectroscopy. Keywords: Logarithmic potential, Airy functions, Bessel functions, Carbon nanotubes, Excitonic transitions, Quantum simulation License: Creative Commons Attribution 4.0 International (CC BY 4.0) Funding: No external funding References: 40 references included in the manuscript