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