Energy level crossings in quantum mechanics

From the eigenvalue equationH_n() =E_n()_n() withH H₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE_n () and the matrix elementsV_mn() where is the independent variable. To solve the dynamical system we need the initial valuesE_n( = 0) and _n( = 0). Thus one finds the motion of the energy levelsE_n(). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H₀ +V₁ +²V₂ and give an application to a supersymmetric Hamiltonian.