Anti-PT Transformations and Complex PT -Symmetric Superpartners

A quantum mechanical system with unbroken superand parity-time (PT )-symmetry is derived and analyzed. Here, we propose a new formalism to construct the complex PT -symmetric superpartners by extending the additive shape invariant potentials to the complex domain. The probabilistic interpretation of a PT -symmetric quantum theory is correlated with the calculation of a new linear operator called the C operator, instead of complex conjugation in conventional quantum mechanics. At the present work, we introduce an anti-PT (APT ) conjugation to redefine a new version of the inner product without any additional considerations. This PT -supersymmetric quantum mechanics, satisfies essential requirements such as completeness, orthonormality as well as probabilistic interpretation.


II. PT -SYMMETRIC SQUARE-WELL SUPERPARTNERS
The solutions of time independent of Schrödinger equation for ground state of the simplest one-dimensional problem in conventional quantum mechanics, i.e., particle in an infinite square-well, is known as, ψ 0c (x) = A sin(kx) ψ 0t (x) = B cos(kx), (1) where k is the wave number ( = 2m = 1). In SUSY QM, the nth-superpotential is defined as the logarithmic derivative of the nth ground state wave function [14], where, n = 1, 2, 3, .... Therefore, the superpotentials for the wave functions of the Eq. (1) are the cotangent (denoted by subscript "c") and the tangent (denoted by subscript "t") functions. Now, to extend the superpotentials to the complex domain, we add an arbitrary imaginary functions, linearly, According to the SUSY QM, the partner potentials are obtained by [14], For n = 1 and due to unbroken SUSY, E (1) 0 = 0, by putting the superpotentials W 1c (x) and W 1t (x) to Eq. (4), we get for cotangent and tangent functions, respectively, and, If the V 1 and V 2 potentials are similar in shape and differ only in the parameters that appear in them, then they are said to shape invariant. The remainder that is defined as Accordingly, partner potentials are shape invariant only if the bracket terms of the remainders to be zero. As a result, the functions f 1c (x) and f 1t (x) are determined by this constraint as, where q is an arbitrary constant. By setting the α = k, the superpotentials are gained as, The real and imaginary terms W 1cr (W 1tr ) and W 1ci ( W 1ti ) are plotted Fig. (1), for typically values q = 2 and k = 1. Proportionally, the complex partner potentials are, and, The potentials V 1 = −1 (q = 0 and k = 1) and real and imaginary parts V 1cr (V 1tr ) and V 1ci ( V 1ti ) are depicted in Fig. (2), and the Fig. (3) illustrates the potentials V 2c (V 2t ) (q = 0 and k = 1) and real and imaginary parts V 2cr (V 2tr ) and V 2ci ( V 2ti ), for typically values q = 2 and k = 1. Finally, The wave functions are obtained by replacing superpotentials (9) in Eq. (2) as, The coefficients A and B are determined by relevant boundary conditions and normalization.

III. ENERGY EIGENVALUES AND EIGENFUNCTIONS
Now consider an infinite square-well in one dimension with length L = π in which a particle moves in the range −π/2 ≤ L ≤ π/2. The boundary conditions, imply that, where n = 0, 1, 2, ... . As a result, the remainders (7) are, and according the unbroken SUSY, E 0 = 0, the energy spectrum is, The Hamiltonian hierarchy yields nth superpotentials, potentials, and eigenfunctions, respectively as follow: V nc (x) = n(n − 1)k 2 n − q 2 csc 2 (k n x) − k 2 n − i(2n − 1)qk n cot(k n x) csc(k n x) V nt (x) = n(n − 1)k 2 n − q 2 sec 2 (k n x) − k 2 n + i(2n − 1)qk n tan(k n x) sec(k n x). and If we set q = 0, these complex quantities become the known real form. So we can interpret that known real potentials are, in fact, particular cases of complex potentials, when the imaginary terms are zero.

IV. PT AND APT TRANSFORMATIONS
The PT -symmetry requires that the potential under the following transformations be symmetric [1]: Consequently, the real V r (x) and imaginary V i (x) terms of the potential must be even and odd functions of x, respectively, Fig. (2) shows that the potentials V 1tr and V 1ti and also Fig. (3) illustrates that V 2tr and V 2ti satisfy these conditions (preserve PT symmetry). These complex PT -symmetric superpartners are isospectrum with infinite square-well (except ground states). Therefore, only the tangent wave functions ψ (n) 0t (kx), corresponding to the bound states. If we shift the width of the well from the range −π/2 ≤ L ≤ π/2 to the range 0 ≤ L ≤ π, we have to replace the tangent with the cotangent functions in order that invariance is preserved under the PT transformations.
However, the superpotential W 1t (x) have anti-symmetric features, in the sense that the real W 1tr (x) and imaginary W 1ti (x) terms of the superpotential are odd and even functions, respectively (see Fig. (1)). Therefore, we can conclude that a PT -symmetric potential is produced by a APT -symmetric superpotential, This is due to the change in the sign of the first spatial derivative, under APT transformations. Moreover, to preserve the potential algebra in SUSY-QM formalism, it is necessary the APT conjugation be similar to the complex conjugation on this operator, The same argument also can be applied to integration. This is because each derivative or integration operation can exchange the odd and even properties of a function. Now, by examining the complex and APT conjugations on the ladder operators, we have, we see that the complex conjugation operator fails past section algebra because it changes the sign of superpotential imaginary term. Therefore, this mathematical operator is not longer applicable for a PT -supersymmetric quantum theory.

V. CONCLUSION
In SUSY QM, a ground state wave function can be written in a general form [14], where function f n (x) is obtained by integrating superpotential, According to past section arguments, in a PT -supersymmetric system, f n (x) and W n (x) should be PT -and APTsymmetric, respectively, With respect to SUSY QM formalism, we know that supersymmetric eigenfunctions consist of a complete set and are orthonormal. Obviously, the PT -supersymmetric eigenfunctions (20) posse also these properties. However, the coordinate-space inner product needs to redefine in such a way that the complex conjugation is replaced by the APT conjugation as, As a consequence, orthonormality reads, and thus, the normalization constant is obtained by, The APT transformations and symmetry are analyzed for a PT -supersymmetric (both PT and SUSY are unbroken) quantum system. However, the APT transformations are less intuitive than PT so we can not exhibit its effect on individual quantities similar to Eq. (21). In combined situations with imaginary number 'i', a typical function ig(x) is PT -symmetric if be odd, and is APT -symmetric if be even function of x, From Eq. (24) we conclude that the first-order differential operators, named the ladder operators, should be redefined by APT conjugation as, accordingly Hamiltonian superpartners are defined as follows: so that H APT 2 = H 1 . Therefore, the mathematical structure of PT -SUSY QM is as same as SUSY QM, with difference that APT conjugation exchange with complex conjugation.