Bound and scattering states for supersingular potentials
Introduction
The use of the Dirac delta function to model some idealized problems in quantum mechanics has a long tradition and is a well-established subject in the standard literature of quantum mechanics — a recent review of models using the Dirac delta function potential is [1]. The standard approach [2], [3] to solve the problem with the Hamiltonian in one dimension runs by considering the Schrödinger equation in the position space and solving the free problem in the two regions separated by the point where the delta potential is localized (), and then applying the boundary conditions that follow from the analysis of the Schrödinger equation in position space with the delta function potential. In other words, if we denote , the conditions on the wave function , for the case of a delta potential , are and . The Hamiltonian is self-adjoint in the domain of wave functions satisfying the above conditions. It is well-known [2], [3] that bound states for the delta potential system exist only when .
A natural extension of the Dirac delta potential system is where is the derivative of the delta function. However, generalization of the usual procedure used to study for the case of has some subtleties [4], [5], [6], [7], specially when one deals with and functions through delta and delta prime sequences. In [7] a distinction was made between local and non-local potentials, and in this work we are considering the local one. The usual treatment of this problem is to study the self-adjoint extensions of the free Hamiltonian [8], [9], in such a way that the interactions are encoded in the boundary conditions [10] for the Schrödinger equation in the position space.
Another way to deal with problems involving point interactions like and is to realize that in these problems there are discontinuities in the wave function and/or its derivatives. Therefore, in terms like and we have the product of a distribution and a function which may be discontinuous at the point of support of the distribution. Since the usual formulations of the theory of distributions [11], [12] start with the space of continuous test functions, one may wonder how to generalize this space of test functions in order to include discontinuous test functions. This problem was considered by Kurasov [13]. In Kurasov’s theory, due to possible discontinuities at , traditional expressions from the theory of distributions like are replaced by where and denotes the one-sided limits
The boundary conditions that follows [13] from the self-adjointness analysis of the problem involving are given by In fact, Kurasov obtained a more general result involving a four parameter family of interactions, with these additional parameters being interpreted as related to a mass jump and to a gauge field [13], [14], although a different interpretation of this family of interaction was provided in [15]. Here we are not considering these two possible extra parameters, so we are being restricted to parameters and only. In [16], Gadella et al. used the conditions in Eq. (2) in order to study the bound states and the scattering coefficients for .
A possible next step is to study a generalization of Eq. (1) of the form Potentials involving derivatives of the delta function of order higher than one are called supersingular potentials. The interpretation of these kind of potentials is certainly subtle and undoubtedly far from controversies. As noticed above, even the case with only the derivative of the delta function already presents controversies [4], [5], [6], [7]. For higher order derivatives of , to the best of our knowledge, only the potential has been previously studied in [17] using sequences of functions. Nevertheless, we can also think of a potential with a sum of terms involving higher order derivatives of the delta function as an approximation for highly localized potentials [18]. In view of the foregoing, we will take a modest approach in this work, leaving aside all the difficulties inherent in the interpretation of interactions involving derivatives of the delta function, and look to this problem as an attempt to generalize the problems with and .
However the usual approaches to deal with and do not prevent problems from arising in the study of supersingular potentials. First of all, let us remind that is not in general a self adjoint operator because of the terms with . The usual approach based on self adjoint extensions of the free Hamiltonian can handle the terms and through appropriate boundary conditions involving the wave function and its first order derivative, but not for . For this reason, supersingular potentials of the form have been considered in [19] in the context of the self-adjoint extension of the operator , but this approach is not an option for us as our problem necessarily involves the free Hamiltonian . On the other hand, nowadays we know that there are interesting and important applications of non-self adjoint operators in Physics. As examples of these applications, see [20], [21] and contributions therein, and [22], [23], [24].
In order to deal with the problem involving , another approach seems to be necessary. One such approach was provided by Lange [25] in terms of an integral version of Schrödinger equation. Boundary conditions for supersingular potential have been obtained from the Schrödinger integral equation, and one thing that stands out in these conditions is that the parameters involved depend on the energy of the solution — unlike, for example, the case seen in Eq. (2). However, no solution for problems with supersingular potentials has been provided in [25].
On the other hand, in [26], [27] the Fourier transform technique was used to solve the fractional Schrödinger equation for delta potentials. Since the usual Schrödinger equation is a particular case of its fractional version, it can also be solved for delta potentials using the Fourier transform. Although the Fourier transform approach apparently seems to be an unnecessary complication with respect to the simpler approach presented in standard textbooks, it allows us to overcome the difficulties linked to the knowledge of the boundary conditions in the physical space. Nevertheless, the solution in physical space, obtained after taking the inverse Fourier transform of this solution in the momentum space, satisfies the boundary conditions. This fact shows that working with the Schrödinger equation in the momentum space is a promising technique to use for solving the problem of the Schrödinger equation with supersingular potentials.
The objective of this paper is therefore to study the one dimensional Schrödinger equation for supersingular potentials, and for this we will use the representation of the Schrödinger equation in the momentum space. Once solutions for bound and scattering states are obtained in the momentum space, using the inverse Fourier transform we can obtain their expression in the position space, and then we can calculate the energy of bound states and the transmission and reflection coefficients of scattering.
In addition to avoiding the issue of boundary conditions, working in the momentum space will allow us to circumvent a very difficult problem which is the product of distributions. Let us assume, for example, the presence of a term in the potential . It is to be expected, by the form of the Schrödinger equation, that its solution in this case will involve a term with , which leads us to the possibility of having to consider a product like . The definition of the product of distributions is a major problem within Schwarz’s theory [11], [12], and this has led to the study of other formulations of theory of generalized functions including the Dirac delta function where it could be possible to define the product of generalized functions [28], such as Colombeau’s theory [29], [30] (but which in turn is not a linear theory like Schwarz’s). Given this, it is certainly desirable to have a formulation for the problem where we can avoid discussions about the product of distributions, which is undoubtedly very important, but totally outside our purposes. Strictly speaking, this does not mean that the problem is intractable if we formulate it in the position space, but only that we still do not know how to deal with it. However, even if it is not possible to treat the problem in the position space, this should not surprise us. Indeed take classic mechanics and its Newtonian, Lagrangian and Hamiltonian formulations as an example. Often one problem is much more difficult to solve in one formalism than in another, and this difficulty can be so great that it is practically intractable with that formalism. Furthermore, even within quantum mechanics, there are controversies about the equivalence of Schrodinger and Heisenberg pictures, as discussed in [31], [32], which could lead to problems that could be dealt within Heisenberg picture but not within Schrodinger picture. We have therefore situations where problems can be dealt with one formalism but not with another, even though these formalisms are considered equivalent.
We organized this paper as follows. In Section 2 we discuss our approach to the problem of solving Schrödinger equation in the momentum space for supersingular potentials for the cases of bound states and scattering states. In Section 3 we outline in detail the solution of the problem of bound states for supersingular potentials, providing an equation for its energy and an expression for its wave function in the position space. We show how our approach reproduces the known cases and discuss new ones. In Section 4 we turn our attention to scattering states, providing an expression for the wave function in the position space and calculating the reflection and transmission coefficients for the scattering by a supersingular potential. We also consider some cases to illustrate our approach. Finally, in Section 5 we present our conclusions.
Section snippets
The Schrödinger equation for supersingular potentials
Let us start with the time independent Schrödinger equation in the momentum space for , that is, where is the Fourier transform of , that is, is the Fourier transform of , that is, recalling that the Fourier transform of is [11], [12], [33], we have and is the convolution which gives
Bound states for supersingular potentials
The solution of the Schrödinger equation for supersingular potentials corresponding to bound states is given by Eq. (5). However, from the expression for in Eq. (4) we see that depends on , and therefore we need to find all these values in order for Eq. (5) to give the wave function of the state. Nevertheless, the values of can be related to and , and therefore the problem ultimately depends on the calculation of and . Non-trivial solutions for
Scattering states for supersingular potentials
The solution of the Schrödinger equation for a scattering state with energy () is given by Eq. (8), where, like in the case of bound states, depends on , and therefore we need to find all these values in order for Eq. (5) to give the wave function of the state. The expression for will be different from the one for bound states because Eqs. (5), (8) are different. We also need to use Eq. (10) instead of Eq. (7). Except for these small differences, the procedure
Conclusions
We have studied solutions of the Schrödinger equation for supersingular potentials, that is, for potentials of the form . The usual approach for this problem in the position space, that is, to study the self-adjoint extensions of the free Hamiltonian, does not work for because the free Hamiltonian involves a second order derivative. However, we can deal with this problem in the momentum space. We solve therefore the Schrödinger equation in momentum space, obtaining the
CRediT authorship contribution statement
S. Jarosz: Conceptualization, Methodology, Writing – original draft. J. Vaz Jr.: Conceptualization, Methodology, Writing – original draft, Validation, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
SJ is grateful to CNPq, Brazil for financial support (process number 142416/2017-7).
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