Joseph Amal Nathan

Spectral singularity and reflectionlessness in non-Hermitian (complex) Ginocchio potential

Recently, interest in scattering from complex non-Hermitian potentials has been renewed both theoretically and experimentally. When a potential becomes complex, several new features arise which may not be predicted. In this regard, we study the non-Hermitian version of the exactly solvable, versatile Ginocchio potential. We find that when it is non-Hermitian and non-absorptive there exists, at most, one spectral singularity (SS). The well-known reflectionlessness of this potential gives way to deep minima in reflectivity. We find the co-existence of an SS with deep oscillations in reflectivity. Remarkably, the complex PT-symmetric version of this potential does not entail any SS. In this case, the reflectionlessness at discrete energies persists with the handedness of reflectivity and non-unitarity. In the new parlance this would be called the invisibility of an optical potential but in this case, uniquely, it is from both sides, left and right.Abstract 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.

Accidental crossings of eigenvalues in the one-dimensional complex PT-symmetric Scarf-II potential

Abstract So far, the well known two branches of real discrete spectrum of complex PT-symmetric Scarf-II potential are kept isolated. Here, we suggest that these two need to be brought together as doublets: E ± n ( λ ) with n = 0 , 1 , 2 …  . Then if strength ( λ ) of the imaginary part of the potential is varied smoothly some pairs of real eigenvalue curves can intersect and cross each other at λ = λ ⁎ ; this is unlike one-dimensional Hermitian potentials. However, we show that the corresponding eigenstates at λ = λ ⁎ are identical or linearly dependent denying degeneracy in one dimension, once again. Other pairs of eigenvalue curves coalesce to complex-conjugate pairs completing the scenario of spontaneous breaking of PT-symmetry at λ = λ c . To re-emphasize, sharply at λ = λ ⁎ and λ c , two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.

A new solvable complex PT-symmetric potential

Abstract We propose a new solvable one-dimensional complex PT-symmetric potential as V ( x ) = i g sgn ( x ) | 1 − exp ⁡ ( 2 | x | / a ) | and study the spectrum of H = − d 2 / d x 2 + V ( x ) . For smaller values of a , g 1 , there is a finite number of real discrete eigenvalues. As a and g increase, there exist exceptional points (EPs), g n (for fixed values of a ), causing a scarcity of real discrete eigenvalues, but there exists at least one. We also show these real discrete eigenvalues as poles of reflection coefficient. We find that the energy-eigenstates ψ n ( x ) satisfy (1): PT ψ n ( x ) = 1 ψ n ( x ) and (2): PT ψ E n ( x ) = ψ E n ⁎ ( x ) , for real and complex energy eigenvalues, respectively.

Real Discrete Spectrum of Complex PT-Symmetric Scattering Potentials

We investigate the parametric evolution of the real discrete spectrum of several complex PT symmetric scattering potentials of the type

Divergence of

V(x)=-V_1 F_e(x) + i V_2 F_o(x), V_1>0, F_e(x)>0

\langle {p}^{6}\rangle

by varying

in discontinuous potential wells

V_2

Real discrete spectrum in the complex non-PT-symmetric Scarf II potential

slowly. Here

Transparency of the complex PT-symmetric potentials for coherent injection

e,o

The Irrationality of ex for Nonzero Rational x

stand for even and odd parity and