We study the non-Hermitian quantum mechanics for the discrete system. This paper gives an exact analytic single-particle solution for an
N-site tight-binding chain with two conjugated imaginary potentials $\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}$ at two end sites, which Hamiltonian has parity-time symmetry (
PT symmetry). Based on the Bethe ansatz results, it is found that, in single-particle subspace, this model is comprised of two phases: an unbroken symmetry phase with a purely real energy spectrum in the region $\ensuremath{\gamma}\ensuremath{\prec}{\ensuremath{\gamma}}_{c}$ and a spontaneously broken symmetry phase with $N\ensuremath{-}2$ real and two imaginary eigenvalues in the region $\ensuremath{\gamma}\ensuremath{\succ}{\ensuremath{\gamma}}_{c}$. The behaviors of eigenfunctions and eigenvalues in the vicinity of ${\ensuremath{\gamma}}_{c}$ are investigated. It is shown that the boundary of two phases possesses the characteristics of exceptional point. We also construct the equivalent Hermitian Hamiltonian of the present model in the framework of metric-operator theory. We find out that the equivalent Hermitian Hamiltonian can be written as another bipartite lattice model with real long-range hoppings.
General properties of eigenvectors and eigenvalues for an effective Hamiltonian governing time evolution in a two state subspace of the state space of the total system under consideration are discussed. The Lee, Oehme and Yang (LOY) theory of time evolution in such a subspace and CP-- and CPT--symmetry properties of the LOY effective Hamiltonian are considered. The CPT transformation properties of the exact effective Hamiltonian for two state subspace are discussed also. Using the Khalfin Theorem we show that contrary to the properties of the LOY theory, the diagonal matrix elements of the exact effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for
t>t0 (
t0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. Using the more accurate approximation than the LOY one we show that the interpretation of the tests measuring the difference between the
K0 mass and the
barK0 mass as the CPT--symmetry test is wrong.
General properties of eigenvectors and eigenvalues for an effective Hamiltonian governing time evolution in a two state subspace of the state space of the total system under consideration are discussed. The Lee, Oehme and Yang (LOY) theory of time evolution in such a subspace and CP-- and CPT--symmetry properties of the LOY effective Hamiltonian are considered. The CPT transformation properties of the exact effective Hamiltonian for two state subspace are discussed also. Using the Khalfin Theorem we show that contrary to the properties of the LOY theory, the diagonal matrix elements of the exact effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for
t>t0 (
t0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. Using the more accurate approximation than the LOY one we show that the interpretation of the tests measuring the difference between the
K0 mass and the
Kˉ0 mass as the CPT--symmetry test is wrong.