We study the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the framework of the position-dependent effective mass Dirac equation. The Dirac equation is mapped into the exactly solvable Schrödinger-like equation endowed with position-dependent effective mass that we present a new procedure to solve it. The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function, the imaginary potential can be obtained. The spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials. We also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials.

this article was funded by SCOAP 3 .Applying the Bethe ansatz method, we investigate the Schrödinger equation for the three quasi-exactly solvable double-well potentials, namely, the generalized Manning potential, the Razavy bistable potential, and the hyperbolic Shifman potential. General exact expressions for the energies and the associated wave functions are obtained in terms of the roots of a set of algebraic equations. Also, we solve the same problems using the Lie algebraic approach of quasi-exact solvability through the (2) algebraization and show that the results are the same. The numerical evaluation of the energy spectrum is reported to display explicitly the energy levels splitting.

We introduce a pipeline including multifractal detrended cross-correlation analysis (MF-DXA) modified by either singular value decomposition or the adaptive method to examine the statistical properties of the pulsar timing residual (PT R) induced by a gravitational wave (GW) signal. We propose a new algorithm, the so-called irregular-MF-DXA, to deal with irregular data sampling. Inspired by the quadrupolar nature of the spatial crosscorrelation function of a gravitational wave background, a new cross-correlation function,σ × , derived from irregular-MF-DXA will be introduced. We show that, this measure reveals the quadrupolar signature in the PT Rs induced by stochastic GWB. We propose four strategies based on the y-intercept of fluctuation functions, the generalized Hurst exponent, and the width of the singularity spectrum to determine the dimensionless amplitude and power-law exponent of the characteristic strain spectrum as H c ( f ) ∼ A yr ( f / f yr ) ζ for stochastic GWB. Using the value of Hurst exponent, one can clarify the type of GWs. We apply our pipeline to explore 20 millisecond pulsars observed by Parkes Pulsar Timing Array. The computed scaling exponents confirm that all data are classified into a nonstationary class implying the universality feature. The value of the Hurst exponent is in the range H ∈ [0.56, 0.87]. The q-dependency of the generalized Hurst exponent demonstrates that the observed PT Rs have multifractal behavior, and the source of this multifractality is mainly attributed to the correlation of data which is another universality of the observed datasets. Multifractal analysis of available PT Rs datasets reveals an upper bound on the dimensionless amplitude of the GWB, A yr < 2.0 × 10 −15 .

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