Abstract The spectrum of the Hermitian Hamiltonian 1 2 p 2+ 1 2 m 2x 2+gx 4 ( g>0), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian H= 1 2 p 2+ 1 2 m 2x 2−gx 4 , where the coupling constant g is real and positive, is PT -symmetric. As a consequence, the spectrum of H is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: when g is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian PT -symmetric − gφ 4 quantum field theory for all dimensions 0⩽ D<3 but is not present in the conventional Hermitian gφ 4 field theory.