The stability and energy exchange mechanism of divergent states with real energy

Abstract

The eigenvalue of the hermitic Hamiltonian is real undoubtedly. Actually, The reality can also be guaranteed by the $PT$-symmetry. The hermiticity and the $PT$-symmetric quantum theory both have requirements regarding the boundary condition. There exists a reverse strategy to investigate the quantum problem. Namely, define the eigenvalue as real first, and, meanwhile, open the boundary condition. Then the behaviors of the wave function at the boundary become rich in meaning. This eigenfunction is generally divergent, and the extent and direction of divergence are closely linked to the energy. It was noted that these divergent behaviors can be well described by their energy-space uncertainty relation which is not trivial anymore. The divergent state is unstable and will certainly exchange energy with the outside. The mechanism of energy exchange is just in the energy-space uncertainty relation, which will benefit dynamic simulation, the many-body problem, and so on. There is no distinct dividing line between this kind of divergent unstable state and the convergent stable state. Their relationship is like that of the rational and irrational numbers. In practice, there are distinct advantages of speed and accuracy for the methods based on the laws of divergence.