Born-oppenheimer approximation for some singular systems

In this thesis we try to solve the energy spectrum of heavy particles in three body singular system by using the Born-Oppenheimer approximation. Heavy particles are carrying attractive Dirac-Delta function potential so, when we solve this system we see that the energy spectrum of heavy particles goes like (m/M)^(1/3). It is different from the well-known articles. After this result we look at the three body system from the many body perspective and we verify that our previous results are true. Afterwards we expand many body equations until second order we find second order energy correction to the original energy. This energy goes like (m/M)^(2/3) and we see not only the second order of the potential term but the second order of the kinetic part also makes a contribution to the second order energy. We find new results show that high order energy corrections can be calculated systematically. In addition to these results we look at Born-Oppenheimer for the relativistic three body system. After some calculations we see the first order of the potential goes like linear as in non relativistic case. This result is very important for us because, we know potential is also linear for the heavy quark model so heavy quark model shows the consistency of our results for the relativistic case. Afterwards we look at this problem from the many body perspective and we verify the truth of our results. Born-Oppenheimer approximation says that you can think the wave function of the heavy and light particles as separable from each other while solving this problem we came across with some cross terms by taking the averages of these terms we show such terms are so small and make contributions to the high order energy. As a result of this thesis we prove the consistency of Born-Oppenheimer in one dimension.