Three PT-symmetric Hamiltonians with completely different spectra

We discuss three Hamiltonians, each with a central-field part H₀$H_0$ and a PT-symmetric perturbation igz$igz$. When H₀$H_0$ is the isotropic Harmonic oscillator the spectrum is real for all g$g$ because H$H$ is isospectral to H₀+g²/2$H_0+g^2/2$. When H₀$H_0$ is the Hydrogen atom then infinitely many eigenvalues are complex for all g$g$. If the potential in H₀$H_0$ is linear in the radial variable r$r$ then the spectrum of H$H$ exhibits real eigenvalues for 0<g<g_c$0<g<g_c$ and a PT phase transition at g_c$g_c$.     

Structural and electronic properties of a single C chain doped zigzag BN nanoribbons

The effects of single C-chain on the stability, structural and electronic properties of zigzag BN nanoribbons (ZBNNRs) were investigated by first-principles calculations. C-chain was expected to dope at B-edge for all the ribbon widths _Nz$N_z$ considered. The band gaps of C-chain doped _Nz$N_z$-ZBNNR are narrower than that of perfect ZBNNR due to new localized states induced by C-chain. The band gaps of _Nz$N_z$-ZBNNR-C(n  ) are direct except for the case of C-chain position n=2$n=2$. Band gaps of BN nanoribbons are tunable by C-chain and its position n, which may endow the potential applications of BNNR in electronics.