Analytical calculations of four-neutrino oscillations in matter

We analytically derive transition probabilities for four-neutrino oscillations in matter. The time evolution operator giving the neutrino oscillations is expressed by the finite sum of terms up to the third power of the Hamiltonian in a matrix form, using the Cayley-Hamilton theorem. The result of computation for the probabilities in some mass patterns tells us that it is realistically difficult to observe the resonance between one of three active neutrinos and the fourth (sterile) neutrino near the earth, even if the fourth neutrino exists.