Darboux Transformation in The Tangent-Squared Potential and Supersymmetry

In this paper, we consider one-dimensional Schrödinger equation for the trigonometric tangent-squared potential. Here, we construct the first-order Darboux transformation and the real valued condition of transformed potential for trigonometric tangent-squared potential equation. Also we obtain the transformed of potential and wave function. Finally, we discuss the correspondence between Darboux transformation and supersymmetry. In order to have supersymmetry and commutative and anti-commutative algebra, we obtain some condition for the corresponding equation.


Trigonometric Tangent-Squared Potential.
Recently, Taseli [22] studied the squared tangent potential V (x) = ν(ν − 1)tan 2 (x), x ∈ ( −π 2 , π 2 ) also known as the symmetric Pöschl-Teller potential [23,24]. It was noted that the energy eigenvalues, but not the eigenfunctions, of the Hamiltonian with the squared tangent potential on the symmetric interval ( −π 2 , π 2 ) are precisely the same as those of the Hamiltonian with the squared cotangent potential V (x) = ν(ν − 1)cot 2 (x) on the asymmetric interval (0, π) considered by Marmorino in [25] (see also [26]). This observation can be easily verified using the identity tan(x + π 2 ) = − cot(x), (1) In this section, we use the asymptotic iteration method to study the Schrödinger equation with the confining potential contained within an infinite square well with sides at x = πa 2 and V 0 gives an indication of how rapidly the potential increases within the well. In this case, the time-independent Schrödinger equation reads We, however, take a slightly different approach to solve (3) by letting the wave function assume the form we rewrite the equation (3) as, In order to have a known associated Legndere equation we take x = a arcsin(y), and one can obtain, where ω = 1 − a 2 E and µ = a 2 V 0 and α = ( Here also we choose suitable variable as, and have, f ′ (y) = u ′ (y)P (y) + u(y)P ′ (y), and f ′′ (y) = u ′′ (y)P (y) + 2u ′ (y)p ′ (y) + u(y)P ′′ (y).
As we know the associate -Legendre differential equation [27][28][29] will be as, Now we compare equations (10) and (11) to each other so one can obtain u(y) as, and So, the corresponding wave function will be as, So, we obtain the energy spectrum and wave function as follow, and where α and β are Legendre parameters and will be constant and n and m are degree of such polynomial.

Darboux Transformation and trigonometric tangent-squared potential
As we know the one-dimensional Schrödinger equation for the trigonometric tangent-squared potential is given by, Now we rewrite the trigonometric tangent-squared potential equation (19) as, Now we take F = (1 − y 2 ), G = −3y and the potential V = ω + µy 2 (1−y 2 ) . Here we can rewrite the above equation as, and In order to have same equation as (19) with different of potential. we have to write following equation, Where V ̸ =V , this lead us to imply C ̸ =Ĉ. In order to obtain the modified potentialV and corresponding wave function for equation (22), we introduce operator ∆ which is called Darboux transformation. The general form of such Durboux transformation will be as, The operator ∆ transforms any solution f (y) of (6) into a new solution.
f (y) = ∆f (y) Where we take special case as A = B. In order to find A or B, we insert the explicit form of the Hamiltonian ∆ and∆ into the Darboux transformation operator (11) and apply it to the solution g(y) of Eq. (6), we can obtain, Making linear independence of g(y) and its partial derivatives, we collect their respective coefficients and equal them to zero, from which we can obtain the following system about the functions A andV .
So, the operator Darboux transformation is Now, we achieve the generalized form of wave function which is corresponding to usual wave functionf (y) as, As we know the usual f (x, t) = e − iEt h f (x) help use to obtain the modified potential, which is given by, ) + 5 (29)

Supersymmetry and Darboux Transformation
In what follows, we will prove that the formalism of supersymmetry for our generalized trigonometric tangentsquared equation is equivalent to the Darboux transformation. Here we suppose such operator i∂ t −θ is self-adjoint, Taking the operation of conjugation on Darboux transformation (23), we obtain where the operator ∆ ⋆ adjoint to ∆ = α(1 − y 2 )(1 + ∂ y ) is given by, Eqs. (21) and (23) can then be rewritten as one single matrix equation of the form, Assuming that H = diag(η,η) and C = (c,ĉ) T , the above equation can be written as, Two supercharge operator Q and Q ⋆ are defined as follows, where ∆ and ∆ ⋆ operators are given by Eqs. (24) and (32), respectively. One can show that the Hamiltonian H satisfies the following expressions Considering the complementing relations of the supersymmetry algebra; the anti commutators {Q, Q ⋆ } and {Q ⋆ , Q}, we obtain the operators R = Q ⋆ Q andR = QQ ⋆ and consider the relations between them with our Hamiltonian η andη. So, one obtain the R andR as follow, where the indices y will be derivative with respect to y. In order to have shape invariance and supersymmetric algebra we need to obtain theR − R. If such value be constant and zero there is some supersymmetry partner for the system. Otherwise we need to apply some condition inR − R to have constant value. So, one can obtain the following equation for theR − RR We mention here that if we want the supersymmetry algebra we need to have also the following commutation relation, and also anti-commutation relation between Q and Q + .
where Q and Q + are supercharges.If we look at the equation (39) we need to apply the conditionR − R to be zero or constant. As we know the limit of x ∈ [− πa 2 , kπa 2 ] we have not shape invariant condition, so we can'nt define supersymmetry. But if we have x ∈ [− kπa 2 , πa 2 ], and k be even we achieve the shape invariance condition and also have supersymmetry charges. For the case of k odd there is not shape invariance. Other case if α (α is constant), or zero we have supersymmetry system.

Conclusion
In this paper we studied the trigonometric tangent-squared equation. We used the first-order Darboux transformation and applied to such system. In order to relate between supersymmetry and Darboux transformation we discussed the supersymmetry algebra and its commutation and anti-commutation superalgebra. We have shown that for satisfying such anti-commutation supercharges theR − R must be constat. Also, we applied the condition on theR − R and shown that the limits of x ∈ [− kπa 2 , kπa 2 ], there is completely guarantee relation between supersymmetry and Darboux transformation, if k be even number.