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Let L be a Schrodinger operator of the form L =-△ + V.After Zhongwei Shens original work in 1995,lots of results related to this kind Schrodinger oprator are developed.In our talk,we will talk about some results we got recently.Using the method of Lharmonic extensions we study the regularity estimates at the scale of adapted BMOα spaces(or Holder spaces).We will also show that a function in BMOα space related to L is the trace of the solution of the Dirichlet problem related to L with this function being the boundary values.This extends the analogous characterization founded by Fabes,Johnson and Neri for the classical BMO space.