Spectral characterization and Schrödinger operator of space-like submanifolds

In this paper, we would like to study space-like submanifolds in a de Sitter spaces S_p^n+p(1). We define and discuss three Schrödinger operators L_H, L_R, L_R/H and obtain some spectral characterizations of totally umbilical space-like submanifolds in terms of the first eigenvalue of the Schrödinger operators L_H, L_R and L_R/H respectively.