J. H. Noble, U. D. Jentschura

The PT-symmetric cubic oscillator with Hamiltonian \(H_3 = – (1/2) d_x^2 + (1/2) x^2 + i G x^3\) is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with \(L^2(R)\) boundary conditions. Eigenfunctions of the stationary Schr\”{o}dinger equation \(H_3 \psi_n(x) = E_n \psi_n(x)\) are manifestly complex, while the energy eigenvalues \(E_n\) are real and positive. Although \(H_3\) does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re \(\psi_n(x)\) and Im \(\psi_n(x)\)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when \(G>0\). Furthermore, we find that the eigenfunctions are “concentrated” in an “allowed” region where the energy \(E_n\) is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex “gain” and “loss” terms mutually compensate and lead to the real energy eigenvalues.

http://arxiv.org/abs/1301.5758

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Atomic Physics (physics.atom-ph)