Supersymmetry Transformations for Delta Potentials

We make a detailed study of the first and second-order SUSY partners of a one-dimensional free Hamiltonian with a singular perturbation proportional to a Dirac delta function. It is shown that the second-order transformations increase the spectral manipulation possibilities offered by the standard first-order supersymmetric quantum mechanics.


Introduction
The study of one-dimensional Hamiltonians with a point interaction has received renewed attention during the past two decades [1,2,3,4,5,6,7]. In general, a point interaction is described by a potential concentrated either in a single or a discrete number of points as it happens, e.g., for the Dirac delta or its derivative. Mathematically, in order to define these potentials, we use the theory of extensions of symmetric operators with equal deficiency indices. These extensions have domains which are characterized by some matching conditions for the wave functions at the points supporting the interaction [2,8,9,10,11]. In particular, the Dirac delta barrier or well have been extensively studied in this way with or without other interactions [12], with or without mass discontinuities at the singular points etc. [13,14].
On the other hand, supersymmetric quantum mechanics (SUSY QM) has emerged as the standard technique for generating new potentials with known spectra departing from an initial one [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]. The method has been applied successfully to regular one-dimensional potentials defined on the full real line [42,43], on the positive semi-axis [44,45] or in a finite interval [46]. Although there are some works dealing with SUSY QM applied to point potentials [47,48,49,50,51,52,53], however the corresponding study has been done just for particular first-order SUSY transformations, without analyzing the full possibilities of spectral manipulation offered by the method. It is interesting to note as well that a point potential may appear as hidden supersymmetries [54,55]. Now, it is the appropriate time for studying the behavior of point potentials with bound states under SUSY QM. Due to the calculation complexity, we shall focus our attention to first and second-order transformations, which anyway are interesting by themselves [56,57,58,59].
We shall restrict the discussion to the following one-dimensional Hamiltonian which is mathematically well defined and self-adjoint provided that we use as its domain D the subspace of the Sobolev space W 2 2 (R/{0}) such that for any ψ(x) ∈ D, one has: where ψ(0+), ψ ′ (0+) and ψ(0−), ψ ′ (0−) are the right and left limits of ψ(x), ψ ′ (x) at the origin respectively [2]. In order to achieve our goal, we have organized this paper as follows: in Section 2, we will study the solutions of the stationary Schrödinger equation for the Hamiltonian H 0 given by (1). In Section 3 we will apply the first-order SUSY techniques, in Section 4 we will analyse the second-order transformations and in Section 5 we will present our conclusions.

Solution of the Schrödinger equation
Let us evaluate in the first place the general solution of the stationary Schrödinger equation for an arbitrary ǫ = −k 2 /2 < 0: with H 0 given in (1). There is one solution vanishing for x → −∞, denoted u + (x), of the form where H(x) is the Heaviside step function, and α, β are constants to be determined from the discontinuity equations (2). We need as well the derivative of u + (x), From equations (4) and (5) it turns out that On the other hand, using equations (2) and (6), we obtain: whereã = a/k. Hence: Inserting these expressions in equations (4) and (5), we finally get Note that the Hamiltonian H 0 in equation (1) is invariant under the change x → −x. Thus, we can find a second linearly independent solution u − (x) for the same ǫ = −k 2 /2, vanishing now for x → ∞, by applying this transformation to u + (x): Moreover: Finally, the general solution of equation (3) for ǫ = −k 2 /2 < 0 is a linear combination of both (7) and (8) which, up to an unessential constant factor, becomes: where D is a constant. The corresponding derivative is given by: Note that, up to normalization, both solutions u ± (x) lead to the same bound state for k 0 = a: The corresponding eigenvalue becomes which coincides with the result derived in [11].
On the other hand, the scattering states for ǫ = κ 2 /2 > 0 can be simply obtained from the solutions given in equations (7), (8) by the substitution k → −iκ, κ > 0. In particular, for a probability flux approaching the singularity from −∞ the corresponding scattering state arises in this way from the u − (x) of equation (8), which (up to unessential constant factor) leads to It is clear now that the reflection R and transmition T coefficients become the standard ones (see, e.g., [60]):

First-order SUSY transformations
Let us start with the initial Schrödinger Hamiltonian H 0 given in (1). As it is well known (see, e.g., [37,41] and the references cited there), its first-order SUSY partner, is intertwined with H 0 in the way where Here, the transformation function u(x) is the seed solution given in (9), associated to the factorization energy ǫ = −k 2 /2 and satisfying equation (3). The SUSY partner potential V 1 (x) of V 0 (x) is given by: We assume the standard restriction ǫ ≤ E 0 ⇒ k ≥ k 0 , in order to avoid the creation of new singularities in V 1 (x) with respect to those of V 0 (x). Note that, from equation (14) and (3) we have: Hence, a straightforward calculation using equations (9), (10) leads to: As ǫ = −k 2 /2, equations (15), (16) give: Note that the denominator of equation (17) never vanishes for x ∈ (−∞, ∞) and D ≥ 0. Moreover, it can be seen that the delta term in V 1 (x) is now repulsive (since a > 0). A straightforward consequence of the intertwining relationship (13) is that for any eigenfunc- Moreover, if ψ satisfies as well equation (2) it turns out that ψ (1) now obeys: which is consistent with the fact that the intensity of the delta term in V 1 (x) has an opposite sign compared with V 0 (x) and the second term of V 1 (x) has just a finite discontinuity at x = 0 (see equation (17)). Concerning the spectrum of H 1 , let us note in the first place that A + 1 transforms the scattering eigenfunctions of H 0 into the corresponding ones of H 1 . In particular, the wavefunction ψ(x) given in equation (11), when transformed by acting on it with A + 1 , produces an expression ψ (1) (x) which is a bit large to be presented here. However, for large values of |x| that expression reduces to the following scattering one (up to a constant factor): This means that the initial reflection and transmission coefficients are unchanged under the first-order SUSY transformation (compare equation (12)). We thus conclude that the continuous spectrum of H 0 belongs as well to the spectrum of H 1 . Let us note that the differences in the spectra of H 1 and H 0 rely in general in the modifications produced by a non-singular SUSY transformation on the discrete part of the initial spectrum. For first-order transformations, these changes can be classified according to the essentially different combinations of the parameters D ≥ 0 and k ≥ k 0 which characterize the seed eigenfunction u(x). We can find three different situations.
(i) Creation of a new ground state at ǫ < E 0 . This case appears for D > 0, k > k 0 . Here, the eigenfunction ψ (ii) Isospectral transformations. These are achieved from the previous case either by taking D → 0 or D → ∞. Since in both situations u(x) goes to zero at one of the ends of the x-domain, it turns out that ψ (iii) Deleting E 0 . This situation arises from the previous one by taking k = k 0 = a (ã = a/k = 1). Since is not normalizable, and then Sp(H 1 ) = [0, ∞). From equation (17), it is clear that now This means that, by deleting the bound state of the attractive delta well V 0 (x) = −aδ(x), a > 0, which is placed at E 0 = −a 2 /2, we recover the repulsive delta barrier of equation (18), a standard result well known in the literature.

Second-order SUSY transformations
In this section it will be illustrated, by means of the delta-well potential, the advantages for manipulating spectra of the second-order SUSY transformations [56,57,58,59] compared with the first-order ones. It is nowadays known that the second-order SUSY partners H 2 of the initial Hamiltonian H 0 can be generated either by employing two eigenfunctions u 1 (x), u 2 (x) of H 0 , not necessarily physical, associated to two different factorization energies ǫ 1,2 , ǫ 1 = ǫ 2 [37,41] or by an appropriate eigenfunction u 1 (x) in the limit when ǫ 2 → ǫ 1 (the so called confluent case [61,62]). In both situations the two Hamiltonians H 0 , H 2 are intertwined by a secondorder operator in the way the new Hamiltonian H 2 takes the standard Schrödinger form and the second-order SUSY partner V 2 (x) of the initial potential V 0 (x) is given by the real function w(x) being proportional in general to the Wronskian of two generalized eigenfunctions of H 0 [63]. An explicit classification of the several second-order SUSY transformations is next given.
4.1 Confluent case [61,62] Let us consider in the first place the limit ǫ 2 → ǫ 1 ≡ ǫ = −k 2 /2 < 0, taking as seed the Schrödinger solution u + (x) vanishing as x −→ −∞, which means to take the u(x) given in equation (9) with D = 0, namely: In this case the real function w(x) appearing in equation (19) takes the form [62] w(x) : An explicit calculation for x ≤ 0 leads to: On the other hand, for x > 0 it turns out that: By combining these two results, we obtain The second-order SUSY partner potential of V 0 (x) becomes now Note that, since u ′ (0+) = k − 2a and u ′ (0−) = k, then u ′ (x) and consequently the potential difference ∆V (x) = V 2 (x) − V 0 (x) have a finite discontinuity at x = 0. In order to avoid the arising of extra singularities for V 2 (x) with respect to V 0 (x) we have to take w 0 ≥ 0. Concerning the spectrum of H 2 , a similar calculation as in the first-order case shows that the scattering eigenfunctions of H 0 are mapped into the corresponding ones of H 2 , i.e., the energy interval [0, ∞) belongs to Sp(H 2 ). As for the discrete part of the spectrum, several possibilities of spectral manipulation emerge, according to how we choose k and w 0 .
(i) Creating a new bound state at ǫ = E 0 . This case appears by taking w 0 > 0 and k = k 0 = a. Since the eigenfunction ψ (2) ǫ of H 2 associated to ǫ is square-integrable, i.e., a new bound state has been created at ǫ, either below the ground state for k > k 0 or above it for k < k 0 . The last option is illustrated in Fig. 1, where we have plotted the potential difference ∆V (x) as a function of x for a = 2, k = 1, w 0 = 1, i.e., a new level was created at ǫ = −1/2 > E 0 = −2 (see the two gray horizontal lines in the same graph). Note the existence of a finite discontinuity in ∆V (x) at x = 0, induced by a similar discontinuity of u ′ (x) at the same point. (ii) Isospectral transformations. They arise in the first place as a limit of the previous case for ǫ = E 0 and w 0 → 0. Note that the long explicit expression for the V 2 (x) of (22) which would appear if we would substitute explicitly the u(x) and w(x) of equations (20), (21) becomes strongly simplified in this limit: Since now An alternative way to produce isospectral transformations is to use the single bound state of H 0 for evaluating w(x). The corresponding formula is achieved from equation (21) by taking k = k 0 = a,ã = 1, which leads to: Hence Note that now w(x) does not have any node for Moreover, in this domain it turns out that i.e., ψ (iii) Deleting the ground state of H 0 . By taking now the limit of equation (23) for w 0 → 0 or w 0 → −1/a, it turns out that lim In both cases ψ (2) ǫ is not square-integrable and then This result means that we have deleted the ground state of H 0 in order to obtain H 2 . For w 0 → 0 the potential of equation (24) becomes On the other hand, for w 0 → −1/a the corresponding potential V 2 (x) is obtained from the previous one by the change x → −x.
Let us remark that, although the final spectra of the SUSY partner Hamiltonians of H 0 are the same when deleting its ground state in the first-order and in the confluent secondorder transformations, however the potentials V 1 (x) and V 2 (x) are physically different (compare equations (18) and (25)). In particular, note the opposite signs of the coefficients of the Dirac delta function for both potentials. [64,65,66] Let us assume that k = k R + ik I is complex with k R > 0, k I ∈ R, and suppose that the two involved factorization energies are now given by ǫ = −k 2 /2 andǭ, wherez denotes the complex conjugate of z. Since we need to avoid the arising of extra singularities in the new potential, we will take a Schrödinger seed solution vanishing at one of the ends of the x-domain in the form given in equation (20) with k ∈ C, namely,

Complex case
To compute now the second-order SUSY partner potential V 2 (x), we have to obtain in the first place the Wronskian W (u,ū) and then the real function This calculation is cumbersome but otherwise straightforward, which leads to: Then, V 2 (x) will be given by with u(x),ū(x) and w(x) as given in equations (26) and (27). An illustration of the potential difference ∆V (x) as function of x for a = 2, k = 1/100 + i/10 is given in Fig. 2. Note that these equations become highly simplified if k R = a: Moreover, for the particular choice k R = a we get a more compact expression for the new potential V 2 (x) than for a generic k R that would appear if we would substitute the u(x) and w(x) of equations (26), (27) in equation (28): − 4ak I sin(2k I x)(a 2 + k 2 I e 4ax ) + 8a 2 k 2 I e 2ax ] a 2 − k 2 I e 4ax − 2a 2 e 2ax cos(2k I x) 2 .
Let us remark that, for the general case characterized by equations (26) and (27) as well as the particular ones described by equations (29)-(31), the scattering states of H 0 are mapped into the corresponding ones of H 2 , and the same happens for the bound state. Thus, it turns out that the spectrum of H 2 will be equal to Sp(H 0 ) = {E 0 } ∪ [0, ∞), i.e., the complex second-order SUSY transformations which produce a real final potential are strictly isospectral.

Real case
Let us take now two seed solutions u 1 , u 2 in the form given in equation (9), associated to the pair of real factorization energies ǫ 2 < ǫ 1 ⇒ k 2 > k 1 . Their explicit forms, and the corresponding derivatives, are given by: whereã i = a/k i . Similarly as in the complex case, the calculation of the Wronskian w(x) ≡ W (u 1 , u 2 ) of the two involved Schrödinger seed solutions is once again cumbersome, but a convenient compact expression reads: By employing this equation, it is straightforward to calculate the new potential through: Concerning the spectrum of H 2 , once again the scattering states of H 0 are mapped into the corresponding ones of H 2 . As for the discrete part of the spectrum, several possibilities are worth of study.
(i) Creating two new levels. Let us suppose first that ǫ 1 = E 0 = ǫ 2 . In order that w(x) do not have nodes, the two factorization energies must be placed either both below (for k 2 > k 1 > a) or both above E 0 (for k 1 < k 2 < a). Moreover, according to the chosen ordering ǫ 2 < ǫ 1 , the solution u 2 (x) must have one extra node with respect to u 1 (x) [37]. In the domain k 2 > k 1 > a (ǫ 2 < ǫ 1 < E 0 ) this can be achieved by taking D 2 < 0 and D 1 > 0 while for k 1 < k 2 < a (E 0 < ǫ 2 < ǫ 1 ) it must be taken D 2 > 0 and D 1 < 0. With this choice of parameters, it turns out that the two eigenfunctions of H 2 associated to ǫ 1 and ǫ 2 , ψ i.e., two new levels have been created for H 2 , either both below the ground state of H 0 (for k 2 > k 1 > a) or both above E 0 (for k 1 < k 2 < a). An illustration of the last situation is shown in Fig. 3, where we have plotted the potential difference ∆V (x) for a = 2, k 2 = 1, k 1 = 1/2, D 1 = −1/2, D 2 = 1. As a result of the transformation, two new levels were created above the ground state energy of H 0 at the positions ǫ 2 = −1/2 and ǫ 1 = −1/8 (see the gray horizontal lines at Fig. 3).
(ii) Creating one new level. This case arises from the previous one for D 2 → 0. Now it turns out that ψ ǫ 2 is not square-integrable anymore, meaning that Thus, in order to generate H 2 a new level has been created at ǫ 1 , above E 0 for k 1 < a and below it for k 1 > a.
(v) Deleting the level E 0 . This can be achieved as a limit of the previous case for D 1 → 0.

Conclusions
We have employed the first and second-order supersymmetric quantum mechanics for generating new potentials with modified spectra departing from the delta well potential. The first-order transformation allowed us to change just the ground state energy level, while the second-order transformations enlarged the possibilities of spectral control, including the option of manipulating the excited state levels. On the other hand, it is important to remember that the first-order transformations induced in the new potential a delta term with an opposite sign compared with the initial one (physically the delta term changed from attractive to repulsive). Meanwhile, the second-order transformations generated a delta term with exactly the same sign as the initial one (the attractive nature was preserved under the transformation). These physical differences should be taken into account in the determination of the most appropriate transformation for building a potential model. We can conclude that supersymmetric quantum mechanics is a powerful mathematical tool, which is quite useful for implementing the spectral design in physics.