Zafar Ahmed

Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex -invariant potential

Abstract The discrete eigenvalues of the complex PT -invariant potential V(x)=(−V 1 sech x−iV 2 tanh x) sech x , V1>0, are shown to be only complex-conjugate pairs when |V2|>V1+1/4, and real otherwise. The PT symmetry is spontaneously broken in the former and unbroken in the latter case. Using one more potential we find that when its real part is stronger than its imaginary part, all the eigenvalues are real, and they are mixed otherwise.

Pseudo-Hermiticity of Hamiltonians under imaginary shift of the coordinate: real spectrum of complex potentials

Abstract We propose that the real spectrum and the orthogonality of the states for several known complex potentials of both types, PT -symmetric and non- PT -symmetric, can be understood in terms of currently proposed η-pseudo-Hermiticity (A. Mostafazadeh, math-ph/0107001) of a Hamiltonian, provided the Hermitian linear automorphism, η, is introduced as e−θp which affects an imaginary shift of the coordinate: e−θpxeθp=x+iθ.

Pseudo-Hermiticity of Hamiltonians under gauge-like transformation: real spectrum of non-Hermitian Hamiltonians

Abstract We report that it is the pseudo-Hermiticity of Hamiltonians under the gauge-like transformation that underlies the reality of the spectrum and orthogonality of states for the non-Hermitian Hamiltonians type Hβ=[p+iβν(x)]2/2m+V(x), which could be both PT -symmetric and non- PT -symmetric. Notably, the eigenstates of Hβ, when it is PT -symmetric, are real and do not satisfy the PT -orthogonality condition.

Energy band structure due to a complex, periodic, PT-invariant potential

Abstract Using a complex PT -invariant, periodic potential of Kronig–Penny type, we show the occurrence of the usual energy band structure. Occurrence of a new type of band structure is also revealed wherein the Brillouin zone boundaries (BZBs: kL=0,π) may not be reached and along with band gaps rounded double bands may occur near the BZBs.

Reflectionless potentials and PT symmetry

Large families of Hamiltonians that are non-Hermitian in the conventional sense have been found to have all eigenvalues real, a fact attributed to an unbroken PT symmetry. The corresponding quantum theories possess an unconventional scalar product. The eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane. For a special class of such systems, it is possible to impose the PT -symmetric boundary conditions on the real axis, which lies on the edges of the wedges. The PT symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless.

Handedness of complex PT-symmetric potential barriers

Abstract Generally, when imaginary part of an optical potential is non-symmetric the reflectivity, R ( E ), shows left/right handedness, further if it is not negative-definite the reflection and transmission, T ( E ), coefficients become anomalous in some energy intervals and absorption is indefinite (±). We find that the complex PT-symmetric potential barriers could be exceptional in this regard. They may act effectively like an absorptive potential for any incident energy provided the particle enters from the preferred (absorptive) side.

Zero width resonance (spectral singularity) in a complex PT-symmetric potential

We show that the complex PT-symmetric potential V(x) = −V1 sech2x + iV2 sech xtanh x entails a single zero-width resonance (spectral singularity) when and the positive resonant energy is given as .

New features of scattering from a one-dimensional non-Hermitian (complex) potential

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time reversal: R( − k) ≠ R(k) and T( − k) ≠ T(k), unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that Rleft( − k) = Rright(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 1. We present a new non-Hermitian parametrization of the 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* = α2, β2) either in T(k) or in T( − k), when αβ > 0. Thirdly, when αβ < 0 it possesses one SS in T(k) and the other in T( − k). Fourthly, when the potential becomes PT-symmetric [(α + β) = 0], we obtain T(k) = T( − k), it possesses a unique SS at E = α2 in both T( − k) and T(k). Lastly, for completeness, when α = iγ and β = iδ there are no SS, instead we get two real energies −γ2 and −δ2 of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound-state eigenvalues. We find them as E+M = −(γ − M)2 and E−N = −(δ − N)2; M(N) = 0, 1, 2, … with 0 ⩽ M(N) < γ(δ).

Reciprocity and unitarity in scattering from a non-Hermitian complex PT-symmetric potential

Abstract In 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 some parametric regimes of scattering from a complex PT-symmetric potential. In special cases, we show that this potential can even become invisible ( R = 0 , T = 1 ) remarkably this time from both left and right sides. We also find that this potential in a parametric regime enjoys a pseudo-unitarity of the type: T + R left R right = 1 .

Tunnelling through the Morse barrier

Abstract The exact and the WKB forms of the transmission coefficient of the Morse barrier have been obtained. Comparing the results obtained from the exact and the WKB analysis on different types of potentials, we present a possible criterion to ascertain the performance of the WKB method. Extraction of bound states of a potential from the transmission coefficient of the inverted potential is also suggested.