This paper introduces the Kolmogorov-Arnold Neural Operator (KANO). KANO is a dual-domain neural operator parameterized using both spectral and spatial bases, offering unique symbolic interpretability. KANO overcomes the purely spectral bottleneck of the Fourier Neural Operator (FNO) while maintaining expressive power for general position-dependent dynamics (variable coefficient PDEs). FNO is only practical for spectrally sparse operators and requires rapidly decaying input Fourier tails. KANO generalizes strongly to position-dependent differential operators, whereas FNO fails. On quantum Hamiltonian learning benchmarks, KANO reconstructs the true Hamiltonian from a closed-form symbolic representation accurate to four decimal places in the coefficients, and achieves $\approx 6\times10^{-6}$ state inaccuracy on projective measurement data, significantly outperforming $\approx 1.5\times10^{-2}$ for FNO trained on ideal full wavefunction data.