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## PT Symmetric, Hermitian and P-Self-Adjoint Operators Related to Potentials in PT Quantum Mechanics

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)

## Origin of maximal symmetry breaking in even PT-symmetric lattices

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)