Gravitational lensing effects on neutrino oscillations around a Schwarzschild black hole immersed in perfect fluid dark matter

We investigate neutrino flavor oscillations in the weak gravitational-lensing regime of a static, spherically symmetric black hole surrounded by perfect fluid dark matter (PFDM). Within a covariant quantum-mechanical framework, we derive the oscillation phases for radial and non-radial propagation of ultrarelativistic neutrinos. While the leading radial phase is independent of the metric function, non-radial trajectories acquire impact-parameter-dependent corrections governed by the PFDM parameter $\lambda$. We construct the flavor-transition probability for two coherent lensed paths and analyze the resulting interference in two and three flavor scenarios. Varying $\lambda$ shifts the oscillation extrema and modifies their amplitudes and contrast, with a non-monotonic dependence on the neutrino mass scale, lens mass, flavor channel, and mass ordering. The $P_{\mu \tau }$ channel shows particularly strong sensitivity to the PFDM contribution for the considered configurations. Moreover, the path-interference terms retain information about the absolute neutrino mass scale. These results characterize PFDM-induced modifications of lensed neutrino oscillations and provide a basis for more realistic treatments including decoherence, averaging, detector resolution, and corrections to lens trajectories and magnifications.