Analytical calculations of four-neutrino oscillations in matter

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