Operator-Analytical Study of Singularities Appearing in Quantum Theory

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For the study of quantum physics on the non-Euclidean space, we configured the 1-dim space with a junction, where we supposed many singularities concentrate on the junction. We handled an electron moving in the space with non-relativistic theory. For the electron we investigated the relation of some self-adjoint extensions and corresponding boundary conditions. We found a sufficient condition in order that the boundary condition has a phase factor. As for the property of singularity in quantum field theory, we handled the physical system of a 2-level atom coupled with a 1-mode laser. We study the singularity of the coupling strength which causes the ground state phase transition as one of quantum phase transition. We characterized it with the energy level crossing.