Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential, \(V(x)\), where the spectrum is again discrete and real. Showing that the Hamiltonian, \(p^2/2m+V(x)\), is pseudo-Hermitian could be challenging, if possible.

http://arxiv.org/abs/1411.3231
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

It is known that when two identical waves are injected from left and right on a complex PT-symmetric scattering potential the two-port s-matrix can have uni-modular eigenvalues. If this happens for all energies, there occurs a perfect emission of waves at both ends. We call this phenomenon transparency. Using the versatile PT-Symmetric complex Scarf II potential, we demonstrate analytically that the transparency occurs when the potential has real discrete spectrum i.e., when PT-symmetry is exact(unbroken). Next, we find that exactness of PT-symmetry is only sufficient but not necessary for the transparency. Two other PT-symmetric domains of Scarf II reveal transparency without the PT-symmetry being exact. In these two cases there exist only scattering states. In one case the real part of the potential is a well devoid of real discrete spectrum and in the other real part is a barrier. Other numerically solved models also support our findings.

http://arxiv.org/abs/1410.5530
Quantum Physics (quant-ph)

We study the coherent scattering from complex potentials to find that the coherent perfect absorption (CPA) without lasing is not possible in the PT-symmetric domain as the s-matrix is such that \(|\det S(k)|=1\). We confirm that in the domain of broken PT-symmetry\(|\det S(k)|\) can become indeterminate 0/0 at the spectral singularity (SS), k=k∗, of the potential signifying CPA with lasing at threshold gain. We also find that in the domain of unbroken symmetry (when the potential has real discrete spectrum) neither SS nor CPA can occur. In this, regard, we find that exactly solvable Scarf II potential is the unique model that can exhibit these novel phenomena and their subtleties analytically and explicitly. However, we show that the other numerically solved models also behave similarly.

http://arxiv.org/abs/1404.1679

Quantum Physics (quant-ph)

It is known that the perfect absorption of two identical waves incident on a complex potential from left and right can occur at a fixed real energy and that the time-reversed setting of this system would act as a laser at threshold at the same energy. Here, we argue and show that PT-symmetric potentials are exceptional in this regard which do not allow Coherent Perfect Absorption without lasing as the modulus of the determinant of the \(S\)-matrix, \(|\det S|\), becomes 1, for all positive energies. Next we show that in the parametric regimes where the PT-symmetry is unbroken, the eigenvalues, \(s_\pm\) of \(S\), can become unitary (uni-modular) for all energies. Then the potential becomes coherent perfect emitter on both sides for any energy of coherent injection. We call this property Transparency.

http://arxiv.org/abs/1308.1270
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Complex extension of quantum mechanics and the discovery of pseudo-unitarily invariant random matrix theory has set the stage for a number of applications of these concepts in physics. We briefly review the basic ideas and present applications to problems in statistical mechanics where new results have become possible. We have found it important to mention the precise directions where advances could be made if further results become available.

http://arxiv.org/abs/1302.2701
Quantum Physics (quant-ph)

In non-relativistic quantum scattering, Hermiticity is necessary for both reciprocity and unitarity. Reciprocity means that both reflectivity (R) and transmitivity (T) are insensitive to the direction of incidence of a wave (particle) at a scatterer from left/right. Unitarity means that R+T=1. In scattering from non-Hermitian PT-symmetric structures the (left/right) handedness (non-reciprocity) of reflectivity is known to be essential and unitarity remains elusive so far. Here we present a surprising occurrence of both reciprocity and unitarity in scattering from a complex PT-symmetric potential. In special cases, we show that this potential can even become invisible (R=0, T=1) from both left and right sides. We also find that this optical potential can give rise to a perfect transmission (T=1) this time without both unitarity and reciprocity (of reflectivity).

http://arxiv.org/abs/1207.6896
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We bring out the existence of at most one spectral singularity (SS) and deep multiple minima in the reflectivity of the non-Hermitian (complex) Ginocchio potential. We find a parameter dependent single spectral singularity in this potential provided the imaginary part is emissive (not absorptive). The reflectionlessness of the real Hermitian Ginocchio’s potential at discrete positive energies gives way to deep multiple minima in reflectivity when this potential is perturbed and made non-Hermitian (complex). A novel co-existence of a SS with deep minima in reflectivity is also revealed wherein the first reflectivity zero of the Hermitian case changes to become a SS for the non-Hermitian case.

http://arxiv.org/abs/1207.1979

Quantum Physics (quant-ph)

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time-reversal: \(R(-k)\ne R(k)$ and $T(-k) \ne T(k)\), unless the potentials are real or PT-symmetric. For complex PT-symmetry scattering potentials, we propose that \(R_{left}(-k)=R_{right}(k)\) and \(T(-k)=T(k)\).So far the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most one. We present a new non-Hermitian parametrization of Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies (\(E_*=\alpha^2,\beta^2\)) either in \(T(k)\) or in \(T(-k)\), when \(\alpha\beta>0\). Thirdly, when \(\alpha \beta <0\) it possesses one SS in \(T(k)\) and the other in \(T(-k)\). Lastly, when the potential becomes PT-symmetric \([(\alpha+\beta)=0]\), we get \(T(k)=T(-k)\), it possesses a unique SS at \(E=\alpha^2\) in both \(T(-k)\) and \(T(k)\).

http://arxiv.org/abs/1110.4485
Quantum Physics (quant-ph); Mathematical Physics (math-ph)