In this paper we give applications of M. Arai and J. Uchiyama [2, Theorems 1.1 and 1.2], which treat the sharp estimates of the growth orders of the eigenfunctions of the Schrödinger operators with potentials oscillating violently at infinity. We can generalize the results of G.B. Khosrovshahi, H.A. Levine and L.E. Payne (Trans. Amer. Math. Soc., 253 (1979), 211-228) and H. Kalf and V.K. Kumar (Trans. Amer. Math. Soc., 275 (1983), 215-229). Also we generalize the result of S. Agmon (J. Analyse Math., 23 (1970), 1-25).