SUSY partners and S-matrix poles of the one-dimensional Rosen–Morse II potential

Among the list of one-dimensional solvable Hamiltonians, we find the Hamiltonian with the Rosen–Morse II potential. The first objective is to analyse the scattering matrix corresponding to this potential. We show that it includes a series of poles corresponding to the types of redundant poles or anti-bound poles. In some cases, there are even bound states and this depends on the values of given parameters. Then, we perform different supersymmetric transformations on the original Hamiltonian using either the ground state (for those situations where there are bound states) wave functions, or other solutions that come from anti-bound states or redundant states. We study the properties of these transformations.


Introduction
This is a new contribution to the study of solvable one dimensional potentials from two different points of view.First of all from the properties of the scattering matrix and the analysis of its poles.Second from the perspective of the successive supersymmetric (SUSY) transformations [1] of the original Hamiltonian with respect to different solutions of the Schrödinger equation with energies given by the different poles of the scattering matrix.We shall show that such solutions associated to the S-matrix poles are particularly suitable to build SUSY transformations with many interesting properties.
Here, the object of our study is focused on the Rosen-Morse II potential, originally developed to describe vibration of molecules [2].This potential, and its Hamiltonian, has been studied in previous articles from different points of view [3,4,5,6,7,8].We intend to continue using the Rosen-Morse II potential, in the same spirit we have used with other solvable one dimensional Hamiltonians such as the Pöschl-Teller potential [6,9,10], the hyperbolic step potential [11], or the Morse potential [12].In the latter, we have found an interesting structure in the scattering matrix.It includes a series of bound and anti-bound simple poles and two series of redundant poles [13,14].On each series, the solutions of the Schrödinger matrix with energies located at the poles of S(E) are related via first order differential ladder operators.These type of ladder operators relate the Gamow wave functions for the resonance poles on the scattering matrix for the hyperbolic Pöschl-Teller potential [10].
In the present paper, we make it explicit the scattering matrix structure of the one dimensional Hamiltonian with Rosen-Morse II potential, which can be exactly obtained.We show the existence of redundant poles and anti-bound poles and, depending on the shape of this potential which, at its turn, depends on the values of certain parameters, the possible existence of bound states.Unfortunately, first order ladder differential operators cannot exist in this case; but there are first order "shift operators" responsible for the shape invariance, which are proven to be quite useful.Also, we observe the absence of resonance poles.This discussion is focused in Section 2.
In the second half of this paper, Sections 3 and 4, we consider the SUSY transformed Hamiltonians after the original Rosen-Morse II.Here, we use different seed functions.The simplest situation is derived from the use of the ground state of the Rosen-Morse II Hamiltonian as a seed function.Then, a whole sequence of SUSY partner Hamiltonians may be explicitly derived.If we depart from an original Hamiltonian which has N bound states, for each of the successive SUSY transformation performed, we loose one bound state.This process stops at the N -th transformation.The resulting Hamiltonian and all Hamiltonians resulting after successive transformations does not have bound states.Nevertheless, the number of redundant poles, which is finite, and the number of anti-bound poles, which is infinite, remain the same.
There are two other possibilities of using other types of seed functions.Should we use one wave function for a redundant state as seed function, the number of poles of S(E) of each kind would be fixed.If this seed function were the wave function for an anti-bound pole (anti-bound state), then a new bound state appears for each transformation.Then, we can say that the anti-bound state becomes a bound state under a SUSY transformation.In any case, the number of redundant poles remains the same.
We finish the present manuscript with a discussion on the equivalence of some SUSY transformations on the Rosen-Morse II Hamiltonians, in Section 4 and some concluding remarks on the applications of our study.

The Rosen-Morse II potential
To begin with, let us write the one dimensional Hamiltonian where the potential V λ (x) is given by (we have chosen ℏ 2 /(2m) = 1) with λ and β parameters whose values will be fixed later.Potential (2.1) is known as the Rosen-Morse II potential [1].In the limit β = 0, it becomes the hyperbolic Pöschl-Teller potential.Thus, the parameter β is responsible for the asymmetry of the potential.Note that the transformation β −→ −β is equivalent to the transformation given by the parity operator x −→ −x, so that we may take β > 0 without loss of generality.In addition, there is the reflection symmetry λ → −λ: will be present in all the subsequent discussion.Henceforth, we will assume that λ > 0, with no loss of generality.This potential has different shapes depending on certain relations of the two parameters β and λ, as may be seen in Figure 1 for some particular values.The situations to be considered are three: • The potential has a minimum and, therefore has a potential well shape if 0 < β < λ 2 − 1 4 .This is shown by the black curve in Figure 1.

On the scattering matrix
Next, we intend to obtain scattering features given by the S-matrix poles corresponding to the Hamiltonian pair {H 0 , H λ }, with H 0 = −d 2 /dx 2 and H λ = H 0 + V λ (x).First of all, we solve the Schrödinger equation H λ Ψ(x) = EΨ(x), which is exactly solvable, provided that we perform a series of changes in both the solution function and the independent variable.We begin with the following changes, which include the introduction of two new parameters k and k ′ , which we give the respective meaning of income and outcome moments: This gives the following equation (appropriate to describe Jacobi polynomials for some specific values of the parameters) We are looking for exact solutions of (2.3) for any value of E, λ and β.Observe that from (2.2), This identity has particular importance as we shall see later.Then, on the resulting equation, we perform a new change in the variable given by so that the Schrödinger equation is transformed into the following second order differential equation [15]: This is a hypergeometric equation, for which the solutions are well known [15].Once we have the general solution of (2.6), we obtain the general solution of the original Schrödinger equation by reverting the transformations given in (2.5) and (2.2).This is: where A and B are arbitrary constants.The definition of the linearly independent solutions ψ(x) and ϕ(x) of the Schrödinger equation is clear after (2.7), where we assume that 1 − ik is not a negative integer.For the hypergeometric function with parameters a, b, c and indeterminate z, 2 F 1 (a, b, c; z), we have used the following notation Note that in (2.7), we use z = 1+tanh(x) 2 as indeterminate.
The objective of this discussion is finding the explicit form of the S-matrix, or scattering matrix, in order to find the structure of its analytic continuation to the complex plane.This may be performed either in the k momentum representation or in the energy representation.We shall show that this objective can be realized.In fact, the S-matrix connects the asymptotic form of the solution as x −→ −∞ with the asymptotic form x −→ ∞.The solvability of this problem comes from the knowledge of the asymptotic forms of the hypergeometric function.Then, the asymptotic forms of (2.7) are: • On the limit as x −→ −∞.Note that 2 F 1 (a, b, c; 0) = 1.In this limit the asymptotic form Ψ − (x) of Ψ(x) given in (2.7) becomes (2.9) • On the limit x −→ ∞ the situation looks a bit complicated.Nevertheless, we use here the following relation: which helps us to determine the asymptotic form Ψ + (x) of (2.7) as From (2.9) and (2.11) we obtain the transition matrix, {T ij }, which is given by Thus, we are in the position to give an expression for the scattering S-matrix, relating ingoing (A − , B + ) and outgoing (A + , B − ) amplitudes, as Thus, the scattering matrix can be expressed in terms of the elements of the transition matrix in the following form (2.15) The interesting scattering features come from the poles of the scattering S-matrix (2.15), which are the zeroes of T 22 that could be complex.Taking into account the explicit form of T 22 given in (2.13) and the properties of the Euler Gamma function, Γ(z), we obtain two conditions that determine the zeroes of T 22 .These are (n = 0, 1, 2, . . .): • Condition 1: • Condition 2: (2.17) These are two different conditions that will yield to different spectral properties, which we analyze next.
In this analysis, we are to use the following relation between Jacobi polynomials and the hypergeometric functions (valid only when a be a non-positive integer): Since for both conditions k and k ′ are pure imaginary numbers, we may use a special notation for their imaginary part.Thus, we define as auxiliary parameters.Then, we study the consequences of each condition separately.
Condition 1.It gives discrete values of the energy depending on n as well as on λ, which are (2.19) Note that we have labelled k and k ′ , as well as ν and µ in terms of n and λ.The first identity in (2.19) comes from (2.4) and (2.16), the second after the definitions after (2.18) and the third is the consequence of (2.4) and (2.16) and some algebra.Now, after (2.16) and (2.4), we have that and (2.23) In the search for scattering features, we should note that bound states may only appear when the potential has a well shape with a minimum, i.e., when 0 < β < λ 2 − 1 4 .The well has a finite depth, with only a finite number of bound states [3].Then, some real poles of the scattering matrix expressed in terms of energies are expected to be eigenvalues of the Schrödinger equation with square integrable eigenfunctions.The solutions ψ β is a necessary condition for the existence of bound states (here we always choose λ > 0).In consequence, n max = ⌊λ−1/2− √ β⌋, where ⌊k⌋ denotes the entire part of the real number k.The corresponding wave functions, ψ {1} λ,n , are square integrable.
In conclusion, the bound states are labelled by n, under the conditions that: These poles of S(E) are characterized by the condition A similar argument of that given in the previous paragraph shows that n r = ⌊λ − 1/2 + √ β⌋.Now, the eigenfunctions ϕ λ,n are not square integrable, so that these poles do not give the energies of bound states.The existence of this kind of poles is due to the potential asymmetry and exists for values of n on the interval λ Note that for β = 0 this asymmetry disappears and in such a case, this type of poles does not exist.Wave function solutions of the Schrödinger equation with energies given by redundant poles are called redundant states.
In summary, the redundant states are labelled by n, under the conditions that: • Anti-bound state poles: These poles are located at values of the energy characterized by In the momentum representation k := √ E, these poles of the scattering matrix S(k) are located on the negative side of the imaginary axis.Antibound state poles appear for values of n such that n > λ − 1/2 + √ β.
Wave function solutions of the Schrödinger equation with energies given by antibound state poles are called antibound states or virtual states.
Thus, the antibound states are labelled by n, under the conditions that: • Resonance poles: No resonance poles exists in this model, assuming λ be real.Energy values: (2.25) Thus, we have the following identity, (2.26) Values of in and out momenta:

.28)
Eigenfunctions for the values (2.25) of the energy: (2) Under Condition 2, we may also find redundant and anti-bound state poles for the scattering matrix S(E) but no bound states.

Redundant poles exists for those values of n in the interval
Finally, anti-bound state poles appear for values of n such that and are infinite in number.We see that we have two lists of redundant poles and anti-bound state poles, one for Condition 1 and the other for Condition 2.
In Figure 2, we depict the behaviour of the imaginary parts of k and k ′ for both Conditions 1 and 2 and for given values of λ > 0 and β.For the given value of the parameters, we observe that poles from Condition 1 may have bound, redundant and antibound states, while poles from Condition 2 may only have redundant or antibound states depending on the values of the parameters.

Remark
In a previous article [12], we have studied the scattering matrix, S(k), in momentum representation, for the one-dimensional Morse potential, where all poles of S(k) are simple and lie on the imaginary axis.We have found series of poles including bound, anti-bound and redundant, for which their corresponding solutions are linked in each series by ladder operators.In such a case, the factorization of the Hamiltonian using ladder operators permits to relate states corresponding to neighbouring poles by these ladder operators, which are first order differential operators.A similar construction does not seem to be possible in here.
In [12], it was shown that for the Hamiltonian with the one-dimensional Morse potential there are two first order differential ladder operators.These ladder operators relate either bound-antibound series of states or two series of redundant states of the same Hamiltonian.For the Rosen-Morse II potential, this kind of first order ladder operators does not exist.Ladder operators could be only constructed as differential operators of n-th order, where n is the order of the Jacobi polynomial that appear as a part of the eigenfunctions ϕ {1} λ,n (x) in (2.23) [7].

SUSY transformations
In this second part of the paper, we are discussing three types of SUSY-partners of our original Hamiltonian using three different types of seed functions: ground state, redundant eigenfunctions and anti-bound eigenfunctions.We recall that the last two types of eigenfunctions are solutions of the Schrödinger equation with energies determined by a redundant pole or an anti-bound pole of the scattering matrix S(E).Although these solutions are not square integrable, they can be used to generate SUSY transformations.Let us discuss those three situations.

SUSY with bound states and shape invariance potential
Let us consider a nodeless wave eigenfunction of the Rosen-Morse II Hamiltonian, i.e., a wave function without zeroes.Nodeless eigenfunctions for the Rosen-Morse II potentials have been classified in [5].
In the set of bound states the only nodeless wavefunction is that of the ground state, ϕ {1} λ,0 (x), of H λ with energy E {1} λ,0 .As is customary in the theory of SUSY transformations, we define the superpotential derived from the ground state as The next step is the construction of the first order factor differential operators B ∓ λ given by From its construction, B − λ ϕ {1} λ,0 = 0.It happens that these operators factorize the initial Hamiltonian H λ and its SUSY partner, which in this case is H λ−1 Replacing the expression on the superpotential W λ (x) in H λ , we find the potential in terms of the superpotential, and from (3.3), the potential V λ−1 is given by From these factorizations the following intertwining relations are satisfied for any two consecutive Hamiltonians, {H λ , H λ−1 }: This is the property of shape invariance of the Rosen-Morse II hierarchy H λ+n , n ∈ Z, carried out by the operators B ± λ+n .This means that the ordinary SUSY transformations (by ordinary, we mean those SUSY transformations that make use of the ground state as seed function, which is the case here), gives rise to the original potential with a shift in the parameters, in particular, λ.The shift operator B − λ , changes H λ into H λ−1 having one bound state less.This is a consequence to the fact that the resulting potential is less deep.One obtains the same result after successive applications of the SUSY transform, so that the potential well becomes more and more shallow.Consequently, after a finite number of transformations and on, the resulting potential cannot have bound states.We depict this effect in Figure 3.
An immediate consequence of (3.6) is that B − λ and B + λ connect the eigenfunctions of H λ and H λ−1 as discussed in the formulas just below, where we denote by H λ to the space spanned by all the eigenfunctions H λ with fixed λ.After the properties of the Jacobi polynomials and the hypergeometric functions, in particular their behaviour with respect to differentiation [15], we obtain the following action of the intertwining operators B ± λ on the outgoing wave functions resulting from the Conditions 1 and 2: where n ̸ = 0 in the action of B − λ and n ̸ = −1 for B + λ .For these particular cases (or, in other words, for ground states transformations), we have the following relations: Once we have analyzed the action of successive SUSY transformations using the original potential ground state as seed function, we wish to use another type of seed functions, assuming them nodeless, such as the wave function of a redundant state pole or an anti-bound state.We shall do it in here next.
First of all, we give an overview on the possibilities of finding nodeless solutions to the Schrödinger equation with eigenvalues either redundant or anti-bound poles of the scattering matrix S(E).

Classification of nodeless solutions.
In order to construct 1-SUSY transforms such that the new potentials have more levels than the original one, we need nodeless seed functions, i.e., functions without zeroes on the real axis.Solutions of the Schrödinger equation with energies given by redundant poles (redundant states) or anti-bound poles (anti-bound or virtual states) may be suitable to act as seed functions.We need criteria to know when these functions are nodeless.To this end, following the classification by Quesne in [5] of the nodeless solutions of the Rosen-Morse II Schrödinger equation, there are three types of these nodeless solutions: Type I nodeless solutions.Those solutions Type II nodeless solutions, ϕ II λ,m (x), characterized by Type III nodeless solutions, ϕ III λ,m (x), characterized by In order to obtain nodeless solutions of type I or II, we need to have a large value of the parameter β.
Solutions for redundant poles with the Condition 1 are of either one of these types, while those of type III correspond to anti-bound states with Condition 2. In the latter case, nodeless solutions appear no matter the value of β > 0.
3.3 SUSY transformation with a seed function given by a redundant state.
Redundant poles appear for large values of β, so that we choose high β in this discussion.However, large values of β may result in the existence of very few or none bound states.Let us give a specific example.If we choose λ = 5.4 and β = 6, the Rosen-Morse II Hamiltonian shows three bound states plus one nodeless redundant state.This redundant state is ϕ {1} λ,m (x) with λ = 5.4 and m = 4 and is of type I, according to the classification given in the previous subsection.In the present case (λ = 5.4, m = 4, β = 6), we have the following pair of supersymmetric Hamiltonians: and therefore where we have used the symbol "tilde" in order to underline the difference between the "ordinary" transformation where one makes use of the ground state from this one.The SUSY operator connecting both Hamiltonians is defined as in (3.2) by Then, the operator B − λ annihilates ϕ {1} λ ,m of H λ , while B + λ will annihilate φλ ,m of H: In any case, the energy of the seed function can be shifted so that it be located at zero by substracting E {1} λ, m in (3.18).In Figure 4, we show how the first order transformation affects to Hamiltonians and bound states.(3.19) on the potential on the left that uses a nodeless redundant solution of type I on dotted blue.Note that the transformation (3.20) preserves the number of bound states, since the new one results on a wave function which is not square integrable.This is depicted in dotted blue at the right.
Note that this transformation does not produce a new bound state.In this example, three bound states for H λ result on three bound states for H λ .Instead to a new bound state, we have a wave function which is not square integrable.It is solution of the Schrödinger equation with energy given by the energy of the redundant pole.This wave function is depicted on dotted blue in the right hand side of Figure 4.This resulting wave function corresponds again to a redundant pole.3.4 SUSY transformation with a seed function given by a anti-bound state.
Now, the point of departure is a Rosen-Morse II potential with parameters λ = 2.4 and β = 1.This Hamiltonian has only one bound state.A nodeless solution of the Schrödinger equation of type III gives the wave function for an anti-bound state.We propose as a seed function ϕ {2} λ,m (x) with λ = 2.4 and m = 2, which fulfills all these conditions.This function is given by a Jacobi polynomial of order two, which is nodeless.In this case (λ = 2.4, β = 1, m = 2), we have the following pair of Hamiltonians related by first order transformation: This transformation is depicted in Figure 5.In this case, the new potential after the SUSY transformation acquires a new bound state.
Note that whenever the transformation is produced by a real anti-bound wave function, then one new bound state arises after the transformation.This is a consequence of formula (3.21),where we write the explicit form of the first ground state after the transformation.The function in the denominator of (3.21) is the wave function for the anti-bound state, which diverges exponentially on both sides.Therefore, its inverse is square integrable.This situation does not arise after a redundant state, which is rapidly increasing in one side, while is bounded on the other side, see Figure 4.
It is interesting to show that there are at least two ways to arrive to the Hamiltonian H by means of SUSY transformations.One has been just described above.The point of departure for another one is the Rosen-Morse potential V λ depicted in Figure 3, left.The parameters here were λ = 5.4, β = 1.This Hamiltonian has four bound states, three redundant states and an infinite number of anti-bound states.Starting from its ground state, we perform a second order SUSY transformation, resulting in a Hamiltonian with only two bound states as the first and second excited states in the original Hamiltonian have been erased.These two Hamiltonians under consideration are (see (3.14)): We first construct the second order operator B− λ , taking into account that the intertwining is given in terms of Wronskians (3.25) The operator B+ λ can be defined as the Hermitian conjugate of B− λ .This second order transformation is depicted in Figure 6.A comparison between Figures 5 and 6 show that Ṽ in (3.22) and V in (3.24) are indeed the same Hamiltonian.This is a particular case of a more general situation as shown on the next Section.Figure 6: At the left a Rosen-Morse II potential with λ = 5.4 and β = 1.After a second order SUSY transformation (3.24), we arrive to the situation depicted on the right hand side.Observe that it is the same result than the obtained after the transformation in Figure 5.This second order transformation erases two levels from the original Hamiltonian.

On the equivalence between different SUSY transformations of the
Rosen-Morse II potential for N odd.
The situation described in the previous subsection is rather general as we intend to show along the present Section.We start by a Hamiltonian H λ with one bound state, E λ+N,0 (where (1) corresponds to the order of the SUSY transformation).Therefore, H α,N −1 the energy corresponding to this anti-bound state, and given by its corresponding pole on S(E), the original Hamiltonian and its first order partner are, respectively, H (1)  α = H α − 2 The Hamiltonian H where the Wronskian does not include the functions ϕ α,0 (x) and ϕ α,N −1 (x).After this transformation, two bound states are left from the N + 1 bound states of the original Hamiltonian.The energy and wave function of the ground state are, respectively, In all the aforementioned SUSY transformations of Rosen-Morse II Hamiltonians based on the poles of the S matrix give rise to rational partner potentials [5].Our results show the interest of the states associated to such singularities; they generate SUSY transformations which constitute a complement to the well known cases based only on bound states.We think that further research on applications of S poles in SUSY transformations will explain many other properties, for instance, related with exceptional polynomials [16], the interaction of magnons-skyrmions [17], the dynamics of perturbed nonrotating black holes [18] etc.Other applications to graphene under magnetic fields have been considered in a number of references [19,20,21,22,23,24].

Figure 1 :
Figure 1: Rosen-Morse II potential for distinct cases considered: black (continuous) for potential well, blue (dashing) for step potential, and red (dotted) asymmetric barrier.
{1}λ,n (x) in(2.22)  are never square integrable.For ϕ {1} λ,n (x) in (2.23), we have three different possibilities.These are:• Bound state poles:Bound states correspond to simple poles of the scattering matrix on the energy representation, S(E), with E < 0. Each of these poles gives a solution of the Schrödinger equation with square integrable wave function.In the momentum representation, according to (2.2), k = √ E + 2β, k ′ = √ E − 2β, these poles of the scattering matrix S(k) are located on the positive side of the imaginary axis.In our terminology, bound state poles are characterized by the condition Im(k λ,n ) > 0, Im(k ′ λ,n ) > 0 or, equivalently, µ λ,n > 0, ν λ,n > 0 and are located at the negative energies {E {1} λ,n } nmax n=0 .Taking into account (2.20) and (2.21) and the above condition, the values of n for which we have bound state poles are those satisfying simultaneously the conditions

Condition 2 .
It comes obvious after(2.16)and (2.17) that Condition 2 can be obtained after a change λ −→ −λ from Condition 1.Let us write our results here just for completeness.

Figure 2 :
Figure 2: Plot of ν {1} λ,n and µ {1} λ,n : (left) λ = 4.1, β = 1 and (right) λ = 1.1, β = 10, (Condition 1 in continuous and 2 in dashing curves).In the left plot: when both functions are positive (0 ≤ n < λ − 1/2 − √ β), there are bound states.When both are negative, we are in the region of anti-bound states n > λ − 1/2 + √ β.Region in the middle, λ − 1/2 − √ β < n < λ − 1/2 + √ β corresponds to redundant poles, as one of the parameters is positive and the other negative.On the contrary, observe that values of ν −λ,n and µ −λ,n (Condition 2) are both negative for the chosen values of the parameters in the left plot, showing the presence of antibound states and no other features.In the right plot: for different values of the parameters λ = 1.1 and β = 10 there is no bound states for both Conditions 1 and 2, only redundant and antibound states exist.

Figure 4 :
Figure 4: At the left, a Rosen-Morse II potential with λ = 5.3 and β = 4.At the right the result of a first order SUSY transformation (3.19) on the potential on the left that uses a nodeless redundant solution of type I on dotted blue.Note that the transformation(3.20)preserves the number of bound states, since the new one results on a wave function which is not square integrable.This is depicted in dotted blue at the right.

Figure 5 :
Figure 5: At the left, Rosen-Morse II potential of (3.22) with λ = 2.4 and β = 1 bearing an only bound state (in red).In dotted blue the anti-bound state wave function.Its SUSY transformation appears on the right hand side.It has two bound states, one in red and the new one in dotted blue.The SUSY operators B∓ λ correspond to (3.23).

( 1 )
α has, consequently, two bound states.The energy and the wave function of the ground state are, respectively, and the wave function of the excited state are given, respectively, by E consider the Hamiltonian H α+N and the SUSY transformation of order N − 1, which gives as final result: .21) If we use the values k λ,n and k ′ λ,n given in (2.20) and (2.21) in the functions ϕ(x) and ψ(x) of the general solution (2.7) as well as relation (2.18), we obtain the following solutions indexed by λ and n: Next, we consider the Hamiltonian H λ+N .It is clear that H λ+N can be obtained from H λ by means of the application of N sucesive operators B + , therefore, it has N + 1 bound states with energies: E {1} λ+N,N .Therefore, if we eliminate the N − 1 intermediate levels by a (N −1)order SUSY transformation we will get a Hamiltonian H(N−1) This is proved in the rest of this section.Let us consider a one dimensional Rosen-Morse II Hamiltonian H α := −d 2 /dx 2 + V α (x), and α ∈ (1/2 + √ β, 3/2 + √ β).These Hamiltonians have a unique bound state.However, if we replace α by α+N , where N is a natural number, N = 1, 2, . . ., then the Hamiltonian has N +1 bound states.To begin with, let us consider H α with α ∈ (1/2 + √ β, 3/2 + √ β).Take an anti-bound state, ϕ {2} α,N −1 (x) of type III.Its SUSY transform of first order has two bound states.The situation is as follows: If we denote by E λ has two bound states with energies E