One of the most fundamental axioms in quantum mechanics is that the Hamiltonian H, which specifies the energy levels and time evolution of the system, is Hermitian. This requirement is expressed by the equation:
H = H^{†}
Where the symbol † represents the combined operations of matrix transposition and complex conjugation. This axiom stems from the requirement that the energy of the system be real and the conservation of the probability. Yet, with the development of quantum mechanics, it was discovered that Hermiticity is a necessary but not sufficient condition for these requirements, and systems described by non-Hermitian Hamiltonians can also meet these conditions. So the Hermiticity requirement can be replaced by a less vigorous condition, namely parity-time (PT) symmetry condition:
PTH = HPT
The parity operator P is defined as:
x→-x, p→-p
And the time reversal operator T is defined as:
p→−p, x → x, i → −i
A simplest parametric family of PT -symmetric Hamiltonians can be expressed as [1]:
The nature of the system can be divided into three regions according to the value of ϵ. When ϵ ≤ 0, the system has an infinite number of real eigenvalues and this is called the region of unbroken PT symmetry. While for ϵ<0, the system has complex eigenvalues and is called the region of broken PT symmetry [1].
Fig.1 Energy levels of the system with different ϵ [1]
Reference
[1] C. M. Bender, "Making sense of non-Hermitian Hamiltonians," Rep. Prog. Phys. 70, 947-1018 (2007).