Sanjib Dey, Andreas Fring, Thilagarajah Mathanaranjan
We propose a noncommutative version of the Euclidean Lie algebra \(E_2\). Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.
http://arxiv.org/abs/1407.8097
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Asiri Nanayakkara, Thilagarajah Mathanaranjan
In this paper we show that the non-Hermitian Hamiltonians \(H=p^{2}-gx^{4}+a/x^2\) and the conventional Hermitian Hamiltonians \(h=p^2+4gx^{4}+bx\) (\(a,b\in \mathbb{R}\)) are isospectral if \(a=(b^2-4g\hbar^2)/16g\) and \(a\geq -\hbar^2/4\). This new class includes the equivalent non-Hermitian – Hermitian Hamiltonian pair, \(p^{2}-gx^{4}\) and \(p^{2}+4gx^{4}-2\hbar \sqrt{g}x\), found by Jones and Mateo six years ago as a special case. When \(a=\left(b^{2}-4g\hbar ^{2}\right) /16g\) and \(a<-\hbar^2/4\), although \(h\) and \(H\) are still isospectral, \(b\) is complex and \(h\) is no longer the Hermitian counterpart of \(H\).
http://arxiv.org/abs/1407.4633
Mathematical Physics (math-ph)
Asiri Nanayakkara, Thilagarajah Mathanaranjan
In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd degree polynomial potentials of the form \(V(x)=(ix)^{2N+1}+\beta _{1}x^{2N}+\beta _{2}x^{2N-1}+\cdot \cdot \cdot \cdot \cdot +\beta _{2N}x\) where \(\beta _{k}^{\prime }\)s are real or complex for \(1\leq k\leq 2N\). The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters \(\beta _{1},\beta _{2}….\) and \(\beta _{2N}\) of the potential. Unlike in the even degree polynomial case, the highest order term in the potential is pure imaginary and hence the system is non-Hermitian. Therefore all the integrations have been carried out along a contour enclosing two complex turning points which lies within a wedge in the complex plane. With the help of some examples we demonstrate the accuracy of the method for both real and complex eigenspectra.
http://arxiv.org/abs/1407.0191
Mathematical Physics (math-ph)
Sanjib Dey, Andreas Fring, Thilagarajah Mathanaranjan
We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schroedinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy eigenspectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.
http://arxiv.org/abs/1401.4426
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)