Darboux transformations for Dunkl–Schrödinger equations with energy-dependent potential and position-dependent mass

We construct arbitrary-order Darboux transformations for Schrödinger equations with energy-dependent potential and position-dependent mass within the Dunkl formalism. Our construction is based on a point transformation that interrelates our equations with the standard Schrödinger case. We apply our method to generate several solvable Dunkl–Schrödinger equations.


Introduction
Dunkl operators [9] are commuting differential-difference operators associated with finite reflection groups.As generalizations of partial derivatives, these operators appear in a wide range of mathematical applications, such as Fourier analysis related to root systems [2], intertwining operator and angular momentum algebra [12] [11], integrable Calogero-Moser-Sutherland (CMS) models [24] [10], harmonic analysis [5], the generation of orthogonal polynomials [17], and nonlinear wave equations [18].Another field of application arises in quantum physics when conventional derivatives are replaced by Dunkl operators.This replacement can be done in the momentum operators, such that the associated Hamiltonian generates governing equations that contain modified derivatives and reflection operators.In the simplest case of a one-dimensional nonrelativistic system [6], the reflection operator entering in the Schrödinger equation coincides with the parity operator.If this equation admits solutions that have parity, then for these solutions the parity operator can be replaced by a numerical parameter, which in some cases allows for closed-form construction of the latter solutions.Based on this concept, results have been obtained for a variety of systems, for example the three-dimensional Coulomb problem [15], three-dimensional relativistic and nonrelativistic systems with radial symmetry [19] [20], and the isotropic two-dimensional Dunkl oscillator in the plane [13] [14], just to mention a few.While the latter applications encompass several types of governing equations, in the present work we will focus on the one-dimensional Schrödinger equation that is equipped with a position-dependent mass and an energy-dependent potential.Implementation of these concepts within the Dunkl formalism lead to a straightforward modification of the weighted Hilbert space associated with the Hamiltonian [21], being very similar to the standard formulation without Dunkl operators [23].In order to find cases of our Dunkl-Schrödinger equation that admits closed-form solutions, we adapt point transformations and Darboux transformations to the present scenario.Both of these techniques have been widely applied to nonrelativistic quantum systems with position-dependent mass, for example in the study of finite gap systems [4], the problem of operator ordering in Hamiltonians [16], and the construction of coherent states [8].In contrast to these and related applications, to the best of our knowledge point transformations and Darboux transformations have not been applied yet to Dunkl-Schrödinger equations equipped with both a position-dependent mass and an energy-dependent potential.After collecting some basic facts about the Darboux transformation in section 2, we outline basic quantum theory in section 3 as it applies to systems with position-dependent mass and energy-dependent potentials.Section 4 is devoted to point transformations and their application to a particular system that admits bound states.In section 5 we develop Darboux transformations for the systems under consideration here, and we apply both the standard and the confluent Darboux algorithm to generate solvable Dunkl-Schrödinger systems that support bound states.

Preliminaries: Darboux transformations
In order to make this work self-contained, we will now present a brief summary of basic facts about Darboux transformations.Our starting point is the pair of Schrödinger equations where ǫ denotes the real-valued stationary energy, the functions U , Û stand for the potentials, and Φ, Φ are solutions of their respective equation.In order to establish an interrelation between the two equations ( 1) and ( 2) by means of a Darboux transformation, we distinguish between the standard algorithm and the confluent algorithm.
• Standard algorithm: We choose solutions (called transformation functions) u 1 , u 2 , ..., u n of our initial equation (1), that pertain to the stationary energies (called transformation energies) ǫ 1 , ǫ 2 , ..., ǫ n , respectively.This means that we have Then, the function is a solution of equation ( 2), provided the transformed potential Û satisfies the constraint We refer to (3) as Darboux transformation of order n.
• Confluent algorithm: In contrast to the previous case, we have a single transformation energy ǫ 1 only, and the transformation functions obey the following system The expressions for the transformed solution (3) and potential (4) remain the same as before.
3 The Dunkl-Schrödinger quantum system Our goal for this section is to set up a Hamiltonian for position-dependent mass within the Dunkl formalism.In addition, we will define a domain for our Hamiltonian, such that it renders hermitian within a suitable Hilbert space.To this end, let us first introduce the simplest case of a Dunkl operator [9] in the form where ν is a real number, and R stands for the reflection or parity operator.While in the scenario of constant mass we have ν > −1/2, in the presence of a position-dependent mass admissible values for the parameter ν are determined through the Hilbert space that we will define below.Next, we use our operator (7) to build the Hamiltonian in the form introducing a position-dependent mass m, and an energy-dependent potential V E , both of which are assumed to be sufficiently smooth functions that are defined on the whole real line except possibly at the origin.Let us simplify subsequent calculations by requiring the mass to have parity.More precisely, we set such that an odd mass is represented by µ = −1, while an even mass corresponds to µ = 1.
In the same way we describe the action of the reflection operator on a function Ψ that the Hamiltonian (8) acts on.We set where in the odd case we have δ = −1 and in the even case δ = 1 holds.Now, the Hamiltonian (8) is hermitian in the weighted Hilbert space L 2 w (R), where the weight function w is given by The Dunkl-Schrödinger equation is obtained from our Hamiltonian (8) in the form where Ψ represents a solution to the equation.Upon taking into account our weight function (11) and the energy dependence of our potential, the modified probability density P for a solution Ψ of (12) in L 2 w (R) is given by This defines the associated modified norm as If the potential V E does not depend on the energy, the term in square brackets equals one.In this case we must require such that the term ∼ |x| does not contribute singularities to the integrand.For constant mass we have µ = 1, implying the known restriction ν > −1/2.If the potential V E is energy-dependent, then the restriction (15) can be dropped, since the term in square brackets can contribute singularities or remove them, depending on the behavior of the potential V E .In such a situation we must find restrictions for the parameter ν on a case-by-case basis.This is illustrated in our application section 4.2.Let us now write (12) in expanded form by substituting (7) and by incorporation of (10).This gives If we distinguish the parameter values for δ and µ, we obtain four special cases of this equation.
Let us show the two cases that arise from odd and even position-dependent mass functions.We will study the two remaining cases in detail below.For an odd-parity mass (µ = −1) our equation takes the form If our mass function has even parity (µ = 1), then the Dunkl-Schrödinger equation ( 16) reads A particular case of this equation is the scenario of constant mass.We obtain for m = 1/2: Note that the mass being constant implies µ = 1.

Point transformations
The first method for solving the Dunkl-Schrödinger equation ( 16) consists in a point transformation that takes it into conventional Schrödinger form by gauging away the first-derivative term ∼ Ψ ′ .Appropriate choices for the free parameter functions then allow for the construction of a solvable case.

Transformation to standard form
We apply the following point transformation The coordinate change x = x(y) remains arbitrary for now, except that it must have an inverse.
It is sufficient if the inverse exists on either the negative or the positive axis, since we are considering solutions with parity only.The specfic form of (20) ensures that the coefficient of the term ∼ Ψ ′ vanishes.Substitution of ( 20) renders ( 16) in the form Let us now recall that the conventional Schrödinger equation with energy-dependent potential Hence, for the transformed Dunkl-Schrödinger equation ( 21) to take the form (22), the following condition must be imposed on the potential U E : Note that this condition arises simply by comparing the coefficients of Φ in the two equations ( 21) and (22).In the special case of constant mass m = 1/2 and µ = 1, the condition (23) reads Since our conditions ( 23) and ( 24) contain three and two free parameter functions, respectively, the same function U E can be obtained by different choices for the latter parameter functions.As such, each U E gives rise to a class of Dunkl-Schrödinger systems with energy-dependent potential and position-dependent mass.We will comment further on this property of ( 23) and ( 24) in our application section 5.3.Let us now present examples for the construction of a Dunkl-Schrödinger system with energy-dependent potential by means of point transformations.

Application
Our starting point is the conventional Schrödinger equation ( 1).If we can generate a potential U E that satisfies the condition (23), such that (1) is solvable, then our point transformation (20) allows for the construction of an associated Dunkl-Schrödinger equation that is also solvable.Let us make the following choices for the position-dependent mass and the potential in the Dunkl-Schrödinger system that we want to construct: where p and q stand for arbitrary real numbers.We will use particular cases of our specific mass profile (25) in the subsequent sections.Note that our change of coordinate is defined and invertible on the positive half-axis, which is sufficient for our purposes.Furthermore, we chose the particular change of coordinate ( 27), such that the conventional Schrödinger equation ( 1) solvable.Substitution of (27) into the potential condition (23) yields In order to keep our calculations simple and transparent, let us take a special case of the mass function (25) by applying the parameter setting p = q = 1.Then (28) becomes The corresponding equation (1) then takes the form The general solution to this equation can be given in terms of hypergeometric functions.We have where c 1 and c 2 represent arbitrary constants.Furthermore, 1 F 1 and U stand for the confluent hypergeometric functions of first and second kind, respectively [1].Since we are aiming for the construction of bound states, we must work with a solution that is bounded on the whole positive half-axis.This condition is not satisfied by the function U , such that we can set c 1 = 1 and c 2 = 0 in (30).As a result, we obtain the particular solution Upon reversing our point transformation (20), we obtain the function Ψ in the form It is convenient to rewrite the confluent hypergeometric function by means of an identity [25] as follows Since the point transformation (20) interrelates the Dunkl-Schrödinger equation ( 16) and its conventional counterpart (1), the function (32) must solve (16).Upon plugging the settings ( 25) and ( 26) for p = q = 1 into ( 16), we obtain the latter equation as Note that even if (32) is a solution to the latter equation, it does not automatically solve the initial Dunkl-Schrödinger equation (12), unless it has the correct parity behavior given by (10).In order to investigate this behavior, we note that both the exponential and the confluent hypergeometric function are even, such that the overall behavior depends on the exponent of the monomial term.Let us distinguish the two cases of δ = −1 and δ = 1.In the first of these cases the exponent reads At this point we cannot proceed until we know the admissible values for the parameter ν.
According to section 3, due to the energy dependence of our potential V E , we must find these admissible values from the behavior of the term ∂V E /∂E that enters in (14).Substitution of (27) yields which gives the standard restriction ν > −1/2 because the confluent hypergeometric function is regular at the origin in the cases of bound states that we are interested in.Upon using our restriction in (33), we obtain Since the exponent is an odd number, the function (32) has odd parity, and as such solves the initial Dunkl-Schrödinger equation.Next we consider the case δ = 1, the exponent of the middle term on the right side of (32) takes the form Hence, if δ = 1 and ν ≥ 1/2, the function (32) has even parity, such that it solves our initial Dunkl-Schrödinger equation.If ν < 1/2, then the exponent depends on ν, resulting in the condition for an arbitrary integer k.If we take into account that ν > −1/2 due to (34), there is no solution of (35).Hence, in the present case we have the overall constraint Now, since we want to construct solutions of bound state type, we restrict the stationary energy, such that the confluent hypergeometric function in (32) degenerates to a polynomial.This restriction is given by equating the first argument of the latter function to a nonpositive integer, leading to where n is a nonnegative integer.Let us state examples of our solutions (32) for the parameter setting ν = 1/2, δ = −1 (odd parity), and the parameter n as given in (36).We find The graphs of these solutions are shown in figure 1.Now we state the corresponding solutions for the case of even parity (δ = 1), leaving the remaining parameters unchanged.This gives Graphs of these solutions are displayed in figure 2. Next, we verify normalizability of our solution (32) with respect to the modified norm (14).Taking into account that (34) is a positive function, and implementing the present settings (27) and µ = 1, the probability density and the associated norm read observe that δ does not enter in the norm due to the mass having even parity.The integral (43) exists for (32) under the energy restriction (36), such that the latter solution is normalizable, and thus represents bound states.Figure 3 shows normalized probability densities associated with the solutions (37)-( 39) and ( 40)-(42), respectively.

Darboux transformations
Besides point transformations that we studied in the previous section, a further method for generating solvable equations of Schrödinger form is given by Darboux transformations.These cannot be applied to the Dunkl-Schrödinger equation ( 12) directly due to its form, such that we first need to convert it suitably.More precisely, we will use the point transformation (20) for this task.In addition, we must distinguish between the cases of odd-parity mass and even-parity mass.

Odd-parity mass functions
Before we start the construction of our Darboux transformation, let us point out that odd-parity mass functions must be negative either on the positive or on the negative axis.Even though there are some applications for negative mass functions, to the best of our knowledge they are generally considered non-physical.For this reason we will develop the Darboux transformation, but omit to state an example for it.Now, if the mass has odd parity, we have µ = −1 according to (9), such that the Dunkl-Schrödinger equation takes the form (17) after introduction of the parameter δ in (10).Next, we apply our point transformation (20), such that the resulting equation is given by ( 21) for µ = −1.In explicit form we have Inspection of this equation shows that the parameters ν and δ from the Dunkl operator (7) are not present anymore.In other words, the Dunkl scenario enters in our system entirely by means of the point transformation (20), provided the mass function has odd parity.Now, in order to apply a Darboux transformation to (44), we must determine a transformation parameter that

Applications
In order to illustrate the formalism developed in the previous paragraphs, we will now perform examples of second-order Darboux transformations with the goal of generating solvable Dunkl-Schrödinger equations with an energy-dependent potentials that admits solutions of bound state type.While in the first application the standard algorithm is used, the second application is devoted to the confluent algorithm.Our starting point for both applications is the Dunkl-Schrödinger equation ( 12) that we will now equip with the following settings There are several motivations for choosing a constant mass function.Besides resulting in comparably simple calculations, our constant mass is a particular case of both ( 25) and ( 50) for the settings p = 1/2, q = 0. We will comment on different mass functions at the end of this paragraph.Now, upon substituting (60), the Dunkl-Schrödinger equation ( 12) takes the specific form (19). Furthermore, a constant mass implies even parity, such that we have µ = 1.In the next step we observe that the above choice for our mass function matches (50) if we set p = 1/2 and q = 0.After incorporation of these settings into equation ( 19), we obtain which is a special case of (51).A particular solution to this equation can be given in the form note that L stands for the associated Laguerre function [1].Recall that the function ( 62) is a solution of the Dunkl-Schrödinger equation (12) only if it has parity that matches the correct value of δ.Let us now apply our point transformation (20) that takes (61) into a form that is needed for applying our Darboux transformation.We choose Observe that the left part is obtained from ( 20) by inserting the present settings (60) for µ = 1, p = 1/2 and q = 0. Upon inserting the explicit form (62), the function Φ is given by As such, it is a particular solution of the equation which is a special case of (52).As expected, the Dunkl parameters δ and ν appear as a constant term that is independent of the variable y.Let us follow the definitions from ( 53) and ( 54) by setting These definitions can be implemented in equation ( 65) that takes the Schrödinger-like form (52). Before we proceed with the Darboux transformation, let us recall a remark we made in section 4 about conditions ( 23) and (24).According to the remark, the same function U E can be constructed from various choices of the parameters m, V E and the coordinate change x.Let us now demonstrate this using the U E given in (66), recall that we obtained the latter function through the settings (60).If we replace these settings by and perform our point transformation (63), we obtain the equation We observe that up to a numerical factor, this equation has the same form as its counterpart (65).We can match both forms by formally redefining the parameter ν in (68) by means of new parameters δ and ν as Substitution renders our equation (68) in the form which coincides with (65) up to parameter naming.Hence, the settings (60) for a constant mass system and (67) for a position-dependent mass system yield the same equation.We can therefore consider the results obtained by application of the Darboux transformation in the subsequent paragraphs as valid for both the constant mass case as well as certain positiondependent mass scenarios.We are now ready to perform our Darboux transformation, such that we must distinguish between the standard and the confluent algorithm.

Standard Darboux algorithm
This algorithm requires that we provide two transformation functions u 1 , u 2 that solve (52) for the present settings.We choose these functions as special cases of our soution (64) as In order to keep our calculations transparent, we want these functions to take a relatively simple form.This can be achieved by selecting the parameters ǫ 1 and ǫ 2 as Substitution of these values into our transformation functions (69) gives after simplifying and collecting terms The transformed solution Φ is obtained from the rule (3) for the case n = 2, that is, we have Furthermore, the transformed potential ÛE reads recall that U E is given in (66).We omit to state the explicit form of the functions (72) and (73) due to their length, but we will give particular cases below after having reverted the point transformation.The solution (72) and the potential (73) enter in equation ( 55), which in the next step we map back to Dunkl-Schrödinger form by the inverted version of (63).It is given by As mentioned above, the explicit form of this function is very large, such that we restrict ourselves to stating a special case here for the settings |ν = 5/2, δ = 1, and E = 6.Substitution yields where I stands for the modified Bessel function of the first kind [1].This solution represents a bound state for our transformed system, as we will demonstrate below.Next, we compute the transformed potential VE from (59) by inserting the present settings p = 1/2 and q = 0, along with (73).We obtain Note that VE depends on the energy E. As in case of the transformed solution, we will give a particular potential (76) for E = 4. Evaluation yields Figure 4 shows a graph of this function, along with the initial potential from (60) and other transformed special cases for specific choices of the stationary energy.The solution (74) and potential (76) enter in the transformed Dunkl-Schrödinger equation that is the partner to (61).It reads Let us now recall that the function ( 74) is an admissible solution for this equation only if its parity matches the correct value of δ.Inspection of the explicit form of (74) reveals that parity is determined by a single monomial term.We have where irrelevant multiplicative constants were discarded.The exponent in (78) must be an odd integer if δ = −1, and it must be an even integer for δ = 1.Let us evaluate the exponent for the first case.We find note that in the last step we used the restriction ν > −1/2.In the second case δ = 1, the exponent (78) takes the form Since ν > −1/2, the exponent cannot attain any even integer value for ν < 1/2.Thus, we must have ν ≥ 1/2, in which case the exponent vanishes.Consequently, the correct parity of our transformed solution (74) is established if we choose ν ≥ 1/2.In the next step we will establish normalizability of our solutions (74) for certain values of the energy.To this end, we recall that the modified norm of a solution to the Dunkl-Schrödinger equation ( 12) is given by (14).In order to be meaningful, the integral this modified norm must exist, and the integrand must be nonnegative.Existence of the integral can be achieved by choosing the stationary energy such that the first argument of the Laguerre function in (62) becomes a nonnegative integer.This condition reads where n is a nonnegative integer.Solving with respect to E gives As a consequence of this choice for the stationary energy, the Laguerre function will degenerate to a polynomial, which renders (62) bounded on the whole real line.This property persists under our Darboux transformation, such that the transformed solution (74) is also bounded on the whole real line.This extends to the whole integrand in (14), such that the integral exists.
In addition we have We do not verify these conditions explicitly here due to the length of the involved expressions.Instead we show graphs of normalized probability densities associated with our transformed solution (74) for different parameter values in figure 5. Recall that the probability density for a solution Ψ is given by (13).Inspection of the figure indicates that the probability densities are nonnegative functions.Note that the energy values chosen in the figures 4 and 5 are taken from (79) for n = 0, 1, 2. Since we know now that our transformed solution (74) represents bound states if the stationary energy is chosen from (79), let us present graphs of particular cases.Figure 6 and figure 7 show graphs of the solutions (74) for the odd-parity case and the even-parity case, respectively.Observe that one of the graphs shown in figure 6 pertains to the function (75).

Confluent Darboux algorithm
We go back to our initial equation (65) that we consider to be in the form (52) due to our settings (66).In contrast to its standard counterpart, the confluent algorithm of our Darboux transformations requires transformation functions that solve the system (5), (6).In the present case of a second-order transformation the latter system reads recall that U E is given in (66).In order to solve the system (80), (81), we observe that the first equation is formally the same as (65), such that a solution u 1 can be taken from (64).After inserting the definition of ǫ in (66) and replacing ǫ by ǫ 1 , we find In the present application we choose the parameter ǫ 1 as Upon substitution into (82), we obtain the explicit form In the next step we need to find the remaining transformation function u 2 by solving equation (81), which can be done by applying integral or differential formulas [22].According to these formulas, the simplest way of constructing a particular solution to (81) is given by where as before Φ was taken from (64).We omit to state the explicit form of the resulting transformation function u 2 after application of the derivative.From this point on the confluent algorithm of the Darboux transformation proceeds in the same way as the standard case.Substitution of the functions (83), (83), (64) into (72) and (73) yields the transformed solution and potential, respectively.After reversing our point transformation, we obtain a solution (74) and an associated potential (76) for equation (77).If the latter solution has the correct parity as indicated by the parameter δ, then it solves the transformed Dunkl-Schrödinger equation (48) for the present settings in (60).Due to the parametric derivatives, the transformed solution and potential do not allow for simplification, such that their explicit forms are long and involved.
As a consequence, it becomes difficult to analyze the probability density associated with the solutions in explicit form.For this reason, we restrict ourselves here to show graphs of the transformed potential only, see figure 8.

Concluding remarks
When applying solution-generating methods like point or Darboux transformations, the presence of an energy-dependent potential imposes an additional constraint on the transformed system, in particular when looking for bound states.This constraint arises from the norm that involves the parametric derivative of the initial system's potential.In general, normalizability is not preserved by either transformation.However, if we assume that the coordinate change x = x(y) does not depend on the energy, then we can derive the following relation from ( 23): We observe that if the mass is nonnegative and the coordinate change is real-valued, then normalizability is preserved by the point transformation (20).For the Darboux transformation an analogous statement cannot be obtained in such a simple manner because the transformed potential depends on the transformation functions that can exhibit a complicated dependence on the energy.A further comment on the Darboux transformations and the restriction (50) is in order.This restriction on the mass can be avoided by assigning the parity parameter δ a fixed value of −1 or 1.Since in this case the transformed potential does not depend on this parameter anymore, the restriction (50) is not necessary.On the other hand, the Darboux-transformed equation must then be cast in a form that allows mapping it back to Dunkl-Schrödinger form, and parity of the transformed solutions must be established.