Under a correctly-defined inner product, a PT-symmetric Hamiltonian's eigenfunctions have positive norms and exhibit unitary time evolution, requirements for quantum theories.
[6] A closely related concept is that of pseudo-Hermitian operators, which were considered by physicists Paul Dirac,[7] Wolfgang Pauli,[8] and Tsung-Dao Lee and Gian Carlo Wick.
[9] Pseudo-Hermitian operators were discovered (or rediscovered) almost simultaneously by mathematicians Mark Krein and collaborators[10][11][12][13] as G-Hamiltonian[clarification needed] in the study of linear dynamical systems.
[17][18][19] In 2003, it was proven that in finite dimensions, PT-symmetry is equivalent to pseudo-Hermiticity regardless of diagonalizability,[20] thereby applying to the physically interesting case of non-diagonalizable Hamiltonians at exceptional points.
This indicates that the mechanism of PT-symmetry breaking at exception points, where the Hamiltionian is usually not diagonalizable, is the Krein collision between two eigenmodes with opposite signs of actions.