We investigate the tunneling time of a wave packet propagating through a non-Hermitian potential Vr − iVi in space-fractional quantum mechanics. By applying the stationary phase method, we derive a closed-form expression for the tunneling time for this system. This study presents the first investigation of tunneling time at the interplay of non-Hermitian quantum mechanics and space-fractional quantum mechanics. The variation in tunneling time as the system transitions from a real to a complex potential is analyzed. We demonstrate that the tunneling time exhibits a dependence on the barrier width d in the limit d → ∞, showing the absence of the Hartman effect. A particularly striking feature of our findings is the potential manifestation of the Hartman effect for a specific combination of the absorption component Vi and the Levy index α. This behavior arises from the fact that the presence of the absorption component Vi leads to a monotonic increase in tunneling time with barrier thickness, whereas the Levy index α reduces the tunneling time. The interplay of these contrasting influences facilitates the emergence of the Hartman effect under a specific combination of Vi and the fractional parameter α.
Fractional calculus has become an essential framework in geophysics, optics, and biological systems to capture long-range correlations and anomalous transport. In this article, we extend this framework to explore a particle in a periodic potential, where the Schrodinger equation is extended to its fractional form. This allows us to study how the Levy index
q governs the formation and inversion of energy bands, offering a pathway to engineer new physical behaviors by tuning
q in periodic quantum systems. We solve the fractional Schrodinger equation (FSE) for periodic rectangular potentials of varying height
V0, barrier thickness
L, and well width
W using an imaginary-time evolution algorithm, supplemented by Gaussian process regression. This analysis reveals a qualitative shift in the system's band structure at
q=2, separating into distinct regimes of dispersion for
q>2 and
q<2. For
q>2, the energy bands invert as symmetric minima emerge within the first Brillouin zone and shift from
k=0 toward
k=±π/a with increasing
q. These degenerate minima define a Bloch-momentum qubit, suggesting an analog to valley degrees of freedom in valleytronics. The
q at which inversion completes scales as
q∝V0−0.28±0.05,
q∝L−0.35±0.08, and
q∝W−0.49±0.06 when varying parameters individually, indicating a tunable sensitivity to potential geometry. In contrast, for
q<2, the ground band hardens around
k=0, with a dispersion following
C∣k∣q+E0 near
k=0. This suggests an effective mass of 0 for
1<q<2 at the band's lowest energy state. These results demonstrate that the Levy index serves as a tunable degree of freedom in quantum periodic systems, capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics through effective-mass control.
When two identical (coherent) beams are injected at a semi-infinite non-Hermitian medium from left and right, we show that both reflection
(rL,rR) and transmission
(tL,tR) amplitudes are nonreciprocal. In a parametric domain, there exists spectral singularity (SS) at a real energy
E=E∗=k∗2 and the determinant of the time-reversed two port scattering matrix, i.e., $|det(S(\ensuremath{-}k))|=|{t}_{L}(\ensuremath{-}k){t}_{R}(\ensuremath{-}k)\ensuremath{-}{r}_{L}(\ensuremath{-}k){r}_{R}(\ensuremath{-}k)|$, vanishes sharply at
k=k∗, displaying the phenomenon of coherent perfect absorption (CPA). In the complementary parametric domain, the potential becomes either left or right reflectionless at
E=Ez. We rule out the existence of invisibility despite
rR(Ei)=0 and
tR(Ei)=1 but $T({E}_{i})\ensuremath{\ne}1$, in this avenue. We present two simple exactly solvable models where expressions for
E∗,
Ez,
Ei, and parametric conditions on the potential have been obtained in explicit and simple forms. Earlier, the phenomena of SS and CPA have been found to occur only in the scattering complex potentials which are spatially localized (vanish asymptotically) and have
tL=tR.
The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practical application of SS where it is desired to maintain outgoing scattering amplitude to a specific value. The behavior of the loci of constant scattering amplitude in the neighborhood of SS are studied for a pure imaginary barrier potential
iV,
V∈R+. We show that these loci are elliptical and self-similar to each other in
E−V plane where
E is energy of the wave. The orientations of these ellipses for reflection (
R) and transmission (
T) amplitude around a given SS are same. This is surprising due to the different mathematical forms of
R and
T. In the vicinity of SS,
R≈T.
We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time (
PT) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), at which point the transmission and reflection coefficients are divergent simultaneously. The second is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound state solution. By systematically exploring the parameter space, we establish a universal relationship governing the number of SSs in these potentials. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.
Existence of a spectral singularity (SS) in the spectrum of a Schr\"odinger operator with a non-Hermitian potential requires exact matching of parameters of the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SSs at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs and with SSs of the second order, as well as potentials obeying a given symmetry, say,
PT-symmetric potentials. We illustrate all these opportunities with examples. We also describe splitting of a second-order SS under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.