We analyse some PT-symmetric oscillators with \(T_d\) symmetry that depend on a potential parameter \(g\). We calculate the eigenvalues and eigenfunctions for each irreducible representation and for a range of values of \(g\). Pairs of eigenvalues coalesce at exceptional points \(g_c\); their magnitude roughly decreasing with the magnitude of the eigenvalues. It is difficult to estimate whether there is a phase transition at a nonzero value of g as conjectured in earlier papers. Group theory and perturbation theory enable one to predict whether a given space-time symmetry leads to real eigenvalues for sufficiently small nonzero values of \(g\).

http://arxiv.org/abs/1409.2672
Quantum Physics (quant-ph)

We discuss space-time symmetric Hamiltonian operators of the form \(H=H_{0}+igH^{\prime}\), where \(H_{0}\) is Hermitian and \(g\) real. \(H_0\) is invariant under the unitary operations of a point group \(G\) while \(H^\prime\) is invariant under transformation by elements of a subgroup \(G^\prime\) of \(G\). If \(G\) exhibits irreducible representations of dimension greater than unity, then it is possible that \(H\) has complex eigenvalues for sufficiently small nonzero values of \(g\). In the particular case that \(H\) is parity-time symmetric then it appears to exhibit real eigenvalues for all \(0<g<g_c\), where \(g_{c}\) is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether \(H\) may exhibit real or complex eigenvalues for \(g>0\). We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.

http://arxiv.org/abs/1405.5234
Quantum Physics (quant-ph)

We study both analytically and numerically the spectrum of inhomogeneous strings with \(\mathcal{PT}\)-symmetric density. We discuss an exactly solvable model of \(\mathcal{PT}\)-symmetric string which is isospectral to the uniform string; for more general strings, we calculate exactly the sum rules \(Z(p) \equiv \sum_{n=1}^\infty 1/E_n^p\), with \(p=1,2,\dots\) and find explicit expressions which can be used to obtain bounds on the lowest eigenvalue. A detailed numerical calculation is carried out for two non-solvable models depending on a parameter, obtaining precise estimates of the critical values where pair of real eigenvalues become complex.

http://arxiv.org/abs/1306.1419
Mathematical Physics (math-ph)

We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender’s argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-x plane (Rep. Prog. Phys. 70 (2007) 947).

http://arxiv.org/abs/1209.6357

Quantum Physics (quant-ph)