Geometric Quantization from Curvature-Modularity Correspondence

This work develops a geometric framework in which discrete quantum spectra emerge intrinsically from curvature on a non-compact affine (1, 1)-curve, without external confining potentials. By introducing a curvature-modularity correspondence, the total Gaussian curvature period is paired with the modular invariant given by the completed Eisenstein series, yielding a canonical scaling quantity-the curvature-modularity constant (CCM). This invariant uniquely sets the quantum energy scale, leading to the exact spectral law En = λn|CCM|2, where the dimensionless eigenvalues are geometry-dependent but scale-invariant. The associated Hamiltonian, derived from the Laplace-Beltrami operator with a curvature-induced correction, is shown to be essentially self-adjoint and to possess a finite discrete spectrum alongside a continuous sector. A distinguished operator ordering α = −1/2 is selected by exact supersymmetric factorization, ensuring a normalizable zero-energy ground state and an analytic spectral gap. Furthermore, a modified semiclassical quantization rule is obtained, in which a global geometric phase fixed by the curvature period replaces the conventional Maslov index. High-precision numerical analysis confirms spectral rigidity and validates the theoretical predictions. The results establish curvature and modular structure as fundamental organizers of quantum spectra, with potential applications in curvature-engineered quantum systems.