Two kinds of quantum adiabatic approximation
  A simple proof of quantum adiabatic theorem is provided. Quantum adiabatic approximation is divided into two kinds. For Hamiltonian H(t/T), a relation between the size of the error caused by quantum adiabatic approximation and the parameter T is given. 
Ming-Yong Ye, Xiang-Fa Zhou, Yong-Sheng Zhang, and Guang-Can Guo