Anti-$\mathcal{PT}$ Transformations And Complex Non-Hermitian $\mathcal{PT}$-Symmetric Superpartners
We propose a new algebraic formalism for constructing complex non-Hermitian $\mathcal{PT}$-symmetric superpartners by extending a conventional shape-invariant superpotential into the complex domain. The resulting potential is an unbroken super- and parity-time ($\mathcal{PT}$)-symmetric shape-invariant potential with real energy eigenvalues, maintaining this property for all parameter values. In order to restore the probabilistic interpretation within a true quantum theory, a new inner product called the $\mathcal{CPT}$-inner product is defined in $\mathcal{PT}$-symmetric quantum mechanics, replacing the Dirac Hermitian inner product. In this work, we propose a new version of the inner product called the anti-$\mathcal{PT}$ ($\mathcal{APT}$)-inner product, $\langle A|B\rangle\equiv |A\rangle^{\mathcal{APT}}.|B\rangle$, which replaces the previous versions without any additional considerations. This $\mathcal{PT}$-supersymmetric quantum mechanics framework also allows for the unification of various areas of physics, including classical optics and quantum mechanics. To validate the theory, we present exact solutions for optical waveguides and the quantum tunneling probability, demonstrating excellent agreement with experimental data for the probability of crossing the potential barrier in the $\rm ^{3}H(d,n)^{4}He$ reaction.