Tomas Azizov, Carsten Trunk

In the recent years a generalization \(H=p^2 +x^2(ix)^\epsilon\) of the harmonic oscillator using a complex deformation was investigated, where \(\epsilon\) is a real parameter. Here, we will consider the most simple case: \(\epsilon\) even and \(x\) real. We will give a complete characterization of three different classes of operators associated with the differential expression H: The class of all self-adjoint (Hermitian) operators, the class of all PT symmetric operators and the class of all P-self-adjoint operators. Surprisingly, some of the PT symmetric operators associated to this expression have no resolvent set.

http://arxiv.org/abs/1108.5923

Quantum Physics (quant-ph)

Yogesh N. Joglekar, Jacob L. Barnett

By investigating a parity and time-reversal (PT) symmetric, $N$-site lattice with impurities \(\pm i\gamma\) and hopping amplitudes \(t_0 (t_b)\) for regions outside (between) the impurity locations, we probe the origin of maximal PT-symmetry breaking that occurs when the impurities are nearest neighbors. Through a simple and exact derivation, we prove that the critical impurity strength is equal to the hopping amplitude between the impurities, \(\gamma_c=t_b\), and the simultaneous emergence of \(N\) complex eigenvalues is a robust feature of any PT-symmetric hopping profile. Our results show that the threshold strength \(\gamma_c\) can be widely tuned by a small change in the global profile of the lattice, and thus have experimental implications.

http://arxiv.org/abs/1108.6083

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech)