Paolo Amore, Francisco M. Fernández, Javier Garcia

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)

Paolo Amore, Francisco M. Fernández, Javier Garcia

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)

Francisco M. Fernández

The purpose of this paper is the discussion of a pair of coupled linear oscillators that has recently been proposed as a model of a system of two optical resonators. By means of an algebraic approach we show that the frequencies of the classical and quantum-mechanical interpretations of the optical phenomenon are exactly the same. Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvalues are also real.

http://arxiv.org/abs/1402.4473

Quantum Physics (quant-ph)

Francisco M. Fernández, Javier Garcia

We show that a recently developed semiclassical expansion for the eigenvalues of PT-symmetric oscillators of the form \(V(x)=(ix)^{2N+1}+bix\) does not agree with an earlier WKB expression for \(V(x)=−(ix)^{2N+1}\) the case \(b=0\). The reason is due to the choice of different paths in the complex plane for the calculation of the WKB integrals. We compare the Stokes and anti-Stokes lines that apply to each case for the quintic oscillator and derive a general WKB expression that contains the two earlier ones.

http://arxiv.org/abs/1312.5117

Quantum Physics (quant-ph)

Francisco M. Fernández, Javier Garcia

We analyse several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. We show that PT-symmetric Hamiltonians with point-group symmetry \(C_{2v}\) exhibit complex eigenvalues for all values of a potential parameter. In such cases the PT phase transition takes place at the trivial Hermitian limit.

http://arxiv.org/abs/1309.0808

Quantum Physics (quant-ph)

Francisco M Fernández, Javier Garcia

We discuss a PT-symmetric Hamiltonian with complex eigenvalues. It is based on the dimensionless Schrodinger equation for a particle in a square box with the PT-symmetric potential \(V(x,y)=iaxy\). Perturbation theory clearly shows that some of the eigenvalues are complex for sufficiently small values of \(|a|\). Point-group symmetry proves useful to guess if some of the eigenvalues may already be complex for all values of the coupling constant. We confirm those conclusions by means of an accurate numerical calculation based on the diagonalization method. On the other hand, the Schrodinger equation with the potential \(V(x,y)=iaxy^{2}\) exhibits real eigenvalues for sufficiently small values of \(|a|\). Point group symmetry suggests that PT-symmetry may be broken in the former case and unbroken in the latter one.

http://arxiv.org/abs/1308.6179

Quantum Physics (quant-ph)

Paolo Amore, Francisco M. Fernández, Javier Garcia, German Gutierrez

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)

Francisco M. Fernández

We discuss the construction of real matrix representations of complex-valued PT-symmetric Hamiltonians. We show the limitations of a general recipe presented some time ago and propose a suitable basis set for a class of one-dimensional problems.

http://arxiv.org/abs/1301.7639

Quantum Physics (quant-ph)

Francisco M. Fernandez, Javier Garcia

We consider a class of one-dimensional nonhermitian oscillators and discuss the relationship between the real eigenvalues of PT-symmetric oscillators and the resonances obtained by different authors. We also show the relationship between the strong-coupling expansions for the eigenvalues of those oscillators. Comparison of the results of the complex rotation and the Riccati-Pad\(\’{e}\) methods reveals that the optimal rotation angle converts the oscillator into either a PT-symmetric or an Hermitian one. In addition to the real positive eigenvalues the PT-symmetric oscillators exhibit real positive resonances under different boundary conditions. They can be calculated by means of the straightforward diagonalization method. The Riccati-Pad\(\’e\) method yields not only the resonances of the nonhermitian oscillators but also the eigenvalues of the PT-symmetric ones.

http://arxiv.org/abs/1301.1676

Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Paolo Amore, Francisco M Fernández

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)