Title: 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.