Analytic neutrino oscillation probabilities

In the work, we derive exact analytic expressions for $(3+N)$-flavor neutrino oscillation probabilities in an arbitrary matter potential term of matrix elements and eigenvalues Hamiltonian. With expressions, demonstrate that nonunitary nonstandard interaction scenarios are physically distinct: they satisfy different identities can principle be distinguished experimentally. The implemented a public code NuProbe, tool probing new physics through oscillations.