Compact perturbative expressions for oscillations with sterile neutrinos in matter

We extend a simple and compact method for calculating the three-flavor neutrino oscillation probabilities in uniform matter density to schemes with sterile neutrinos, with favorable features inherited. The only constraint of the extended method is that the scale of the matter potential is not significantly larger than the atmospheric $Δm^2$, which is satisfied by all the running and proposed accelerator oscillation experiments. Degeneracies of the zeroth order eigensystem around solar and atmospheric resonances are resolved. Corrections to the zeroth order results are restricted to no larger than the ratio of the solar to the atmospheric $Δm^2$. The zeroth order expressions are exact in vacuum because all the higher order corrections vanish when the matter potential equals zero. Also, because all the corrections are continuous functions of matter potential, the zeroth order precision is much better than $Δm_{⊙}^{2}/Δm_{atm}^{2}$ for the weak matter effect. Numerical tests are presented to verify the theoretical predictions of the exceptional features. Precision and speed comparisons with previous $3+1$ methods are performed. Moreover, possible applications of the method in experiments to check the existence of sterile neutrinos are discussed.