Abstract
We consider the one-dimensional Dirac equation with the most general relativistic contact interaction supported on two points symmetrically located with respect to the origin. We use a distributional method to determine the shape of the interaction, which, in the present case, is equivalent to the standard method of defining contact interactions by self-adjoint extensions of symmetric operators. The interaction on each of these two points depends on four parameters, each one having a clear physical meaning. We are interested in the scattering and confining properties of this model. We focus our attention on even or odd interactions under parity transformations and investigate the existence of critical and supercritical states, bound states, confinement, and scattering resonances for some particular interactions of special interest.
1 Introduction
One-dimensional point interactions (PI) in quantum mechanics have attracted significant attention, both for their intrinsic mathematical and theoretical appeal and for the flexibility afforded by their 4-parameter characterization [1–5], which enables modeling a wide range of short-range interaction systems. From a mathematical and theoretical point of view, the properties of these interactions have been extensively studied using methods such as regularization [2, 6–9], renormalization [10], self-adjoint extensions (SAEs) [1, 11, 12], and distribution theory [4, 13–17]. PIs also serve as idealized models in numerous physical contexts. Notable examples include their roles in exactly solvable many-body systems such as the Lieb–Liniger model [18, 19] and the Tonks–Girardeau gas [20–23], as well as in investigations of the Casimir effect [24, 25], Y-junctions [26], and other quantum systems with localized interactions. For an overview of recent developments in both the theoretical treatment and practical applications of one-dimensional contact interactions, see [27].
The non-relativistic scattering by two general point interactions has been analyzed in connection with tunneling times [28]. The resonant structure of such systems, which is of particular relevance for junction heterostructures [29–31], has been further investigated in [32, 33] (see also [34]). In [35], the authors examined the one-dimensional Hamiltonian modified by two non-local interactions, employing a renormalization approach to the coupling constant. Relativistic point interactions, on the other hand, may provide corrections to non-relativistic models and are relevant for describing impurities in quasi-one-dimensional graphene nanoribbons [36]. These systems have been investigated by many authors, for example, [37–45]. Similarly to the non-relativistic case, they also constitute a four-parameter family of interactions [46, 47]. In particular, the scattering of a relativistic particle by two-point interactions was originally analyzed by [48], in which the author considered two barriers as given by the appropriate limit of rectangular barriers and obtained resonant tunneling at all energies for certain values of the interaction strength, a result later disproved by [49]. However, a systematic study of relativistic resonant scattering for an arrangement of two-point barriers seems to be lacking in the literature and will be addressed in this work, particularly in the case of arrangements having well-defined parity.
At this point, it may be convenient to recall that scattering resonances are characterized by diverse but equivalent conditions, such as i) pairs of poles of the analytic continuation of the -matrix (either on the energy or momentum representation), ii) singularities of the transmission coefficient, iii) zeroes of the Jost function, or iv) the purely outgoing boundary condition, in which the coefficient of the incoming wave function is taken equal to zero for certain (complex) values of the energy (equivalently, of the momentum). Pairs of values of the energy characterizing scattering resonances are always complex, and the members of each pair are complex conjugates of each other. See [50–56]. We shall discuss the emergence of scattering resonances in our model.
In order to carry out a thorough analysis of the two-point relativistic interaction, we employ the distributional approach (DA) to contact interactions developed in [13, 14, 17, 45]. This framework provides an explicit formulation of the interaction terms and expresses them directly in terms of the physical fields [45] (see also [57]). Such a formulation is particularly well-suited for symmetry analysis.
This article is organized as follows: In Section 2, we recall the most general form of the two-point relativistic contact interactions and their transformation under parity transformations. In Section 3, the form of the wave function and the transmission matrix are introduced, and the single-point interaction limit is discussed. Section 4 is devoted to a general analysis of critical, supercritical, and bound states and scattering resonances, leaving a detailed study of special cases of particular interest for Section 5. The article closes with some concluding remarks in Section 6 in addition to two appendices. In Supplementary Appendix A, we establish the relations between the four physical parameters that fix the distributional form of each of the contact interactions, with those parameters that fix the matching conditions of the wave functions on each point supporting the interactions. Each of these matching conditions gives a self-adjoint determination of the Hamiltonian under study. In Supplementary Appendix B, we give the non-relativistic limit of some selected Dirac equations derived from some of these self-adjoint determinations of the Hamiltonian.
2 Analysis of symmetry under parity transformation
The one-dimensional stationary Dirac equation for a particle of mass and energy scattered by a double barrier of singular interactions can be written as a distributional differential equation of the form (we adopt natural units throughout this article, ) [14, 45].where and , with indicating the Pauli matrices. The distributional interaction represents the most general distribution associated with the singular interaction.
Here we consider a double barrier of singular interactions placed at points . Then, following [45], is given by In this expression, and stand for the one-sided limits of the spinor distribution (which, if they exist, can be defined even at singular points [58, 59]). The physical parameters , , and , with , are the strengths of the scalar, vector, and pseudoscalar point interactions1 located at and , respectively.
The analysis of the properties of the interaction under space reflections, as well as the physical interpretation of the interaction parameters, is more readily done in terms of the physical parameters in Equation 2; thus, this will be the main form adopted in this section. However, the bound states and resonances that will be calculated in the following sections are often expressed more compactly in terms of the familiar -matrix parameters used in SAEs and establishing the boundary conditions for the spinor across the singular points aswith , , given by the well-known expression [14, 47]:where and are (dimensionless) constants. The relationships between the -matrix parameters in Equation 4 and the physical parameters in Equation 2 were explicitly obtained in [45] (see also [44, 57]) and, for convenience, are reproduced in Supplementary Appendix A. The point interaction is said permeable (or penetrable) at if all the -matrix parameters are finite; otherwise, it is said to be impermeable (or impenetrable) at . See the Supplementary Appendix A for the conditions of permeability of a point interaction in terms of the physical parameters.
Because we are interested in interactions with well-defined parity, let us analyze how the interaction term in
Equations 1,2, transforms under a space-reflection transformation
. The transformation properties of the distributional spinor
under
are obtained much in the same way as for the usual (non-distributional) Dirac equation [
60,
61], and it can be shown that in a particular (laboratory) reference frame, we have [
45]
Then, noticing that under space reflection
,
, and
, we obtain
This expression is to be compared with
Equation 2; to determine the specific symmetry of the arrangement of interactions, as below.
- –
Even arrangement
For an interaction to be even under space reflection, the interaction distribution must be invariant; that is, it must have the same formal expression after the transformation:which, comparing Equations 2, 5, can be satisfied only if:
In terms of the
-matrix parameters, indicating by
the matrix at
and by
the matrix at
, the condition above for even interaction can be written, using
Supplementary Appendix A6–A9, as
where the superscript “(e)” stands for
even. Notice that the
individualmatrices
do not necessarily possess any symmetry. What is required is that their
arrangementbe symmetrically even [
45].
- –
Odd arrangements
For an interaction to be odd, it must be such thatwhich can be satisfied only ifand the corresponding odd -matrices giving the boundary conditions at again follow from Supplementary Appendix A6–A9 and are given by
Again, the individual matrices do not necessarily characterize an odd interaction; only their arrangement is odd [45].
3 General eigenstates and 1-point limits
Let us now consider the solution of the Dirac equation with two-point interactions, obtaining expressions for the transmission and reflection amplitudes. The double barrier divides the space into three distinct regions, in each of which the Dirac particle moves freely. Thus, we can write the solution asand, for , we have2,3where , is the free Dirac spinor in momentum space, ( here indicates the transpose), and we introduced the matrix defined by
The coefficients , of the wave function outside the interaction region can be connected via the transfer matrix [62], defined here aswhere
3.1 The limit
We now analyze the limit in which both singular interactions converge to a single point. To this purpose, it is convenient to define a connection matrix, , establishing a relationship between the wave function on both sides of the interaction asthat is
From Equation 20, we see that because in the middle of the above equation. Thus, the composition law in this limit is given by the group structure, the same as in the non-relativistic case [28, 63].
From Equations 10, 15, it follows that the single-barrier limit for even and odd arrangements of double barriers are, respectively,and
The single-barrier limit of an even arrangement of interactions, Equation 21, is an even single-point interaction [14, 45], thus preserving the symmetry of the underlying double interactions, although the limit for odd arrangements of two singular barriers does not necessarily preserve the symmetry. In order to correspond to a single odd interaction, the off-diagonal elements of the -matrix must be zero [45]; from Equation 22, this will occur if, and only if, or (in terms of the physical strengths Supplementary Appendix A1–A4, this condition reads or, respectively). In the first case, , neither of the two-point interactions necessarily has a defined parity; the single point limit is , which corresponds to an effective “singular gauge field” interaction [45] (with the phase depending on all the potentially non-vanishing strengths , , and of the original arrangement of two-point interactions). The second case, , corresponds to an odd arrangement of two general odd-point interactions. The single point limit iswhich corresponds to a mixture of a pseudoscalar and magnetic point fields (both depending on the parameters and of the original two-point interactions). Finally, from Equations 21, 22, it is straightforward to observe that both even arrangements of two odd interactions and odd arrangements of two even-point interactions converge to the free case when .
In summary, in the limit , even arrangements of two-point interactions always converge to a single even-point interaction. On the other hand, odd arrangements will converge to a single odd-point interaction in that limit if, and only if, the two-point interactions are odd themselves (i.e., ) or in Equations 11–14. The last condition includes the case of an odd arrangement of two even-point interactions .4 On the other hand, odd arrangements will converge to a nontrivial even single-point interaction in the limit if, and only if, and in Equation 22; this is an interesting and unexpected result, and it illustrates the nontrivial features of point interactions. In all other cases, the limit of odd arrangements of two-point interactions does not have a well-defined parity.
4 Critical, supercritical, bound, resonant, and scattering states
In this section, we will consider the existence of critical and supercritical ( and , respectively) bound states and resonances for the Dirac equation (Equations 1, 2). Critical (supercritical) states emerge when a bound state is absorbed (emitted) from (to) the continuum spectrum of the allowed energies, by varying the parameters of the interaction. Critical and supercritical states can be bound or quasi-bound states, depending on whether they are normalizable or not [64]. We will investigate resonances as the complex poles of the scattering -matrix, which coincide with the complex energy solutions for purely outgoing scattering states. We will start the section by classifying the states, and then we will study those very same states for symmetric arrangements.
4.1 Classification of states
4.1.1 Critical states
Critical states are solutions of the Dirac equation, with ; that is, they are solutions of
In the three regions separated by the barriers, the Equation 16 assumes the formwhere
The requirement of boundedness of the spinor components as implies . The coefficients and are connected with one another by a relation similar to Equation 18, with the matrix in Equation 19 replaced by (and with ). Thus,
The equation above can be written aswhere is the -element of the matrix . So, the condition for nontrivial solutions is
Thus, the system will admit a nontrivial critical state only if the equation above is satisfied; in this case, the critical state is a quasi-bound state because the spinor is not square integrable (it is a nonzero constant at infinity).
4.1.2 Supercritical states
Supercritical states are solutions of the Dirac equation with energy .
Following the same steps as in the previous section, we find the condition for the existence of nontrivial supercritical states to bewhere is the -element of the matrix :and
Again, supercritical states are quasi-bound states.
4.1.3 Bound states
The conditions for the existence of bound states with follow directly from the formalism developed in Section 3. In this case, in Equation 17, with . The square-integrability of the Dirac spinor requires that because the terms in Equation 17 with these coefficients diverge at . Using the boundary conditions (Equation 3) in Equation 18 leads towhere the matrix is given by Equation 19. Nontrivial solutions of Equation 25 exist if, and only if
4.1.4 Resonances
By imposing boundary conditions for a purely outgoing scattering state on Equation 16, we obtain and, thus, exactly the same Equation 26, but with changed to , where . The real parts of the complex energy solutions are the resonant energies.5
4.1.5 Scattering
For scattering solutions with the particle incident from the left (no incident wave from the right), we can set , , , and , obtainingwhere indicates the element of the transfer matrix (Equation 19); and are the amplitudes of reflection and transmission, respectively.
It is also interesting to write the form of the scattering matrix. If we denote the entries of the transfer matrix by , the form of the scattering -matrix in terms of these entries is given by
Observe that the poles of the scattering -matrix, which are given by the same condition obtained by the purely outgoing conditions (namely, ), coincide with those of the transmission coefficient, Equation 27. Therefore, we use this criterion to construct the graphics giving resonances in the cases to be considered later.
4.2 Symmetric arrangements
Now that we have defined all types of states to be considered, let us investigate the dependence of such states on the parameters of the interaction for some specific arrangements of well-defined symmetry.
4.2.1 Even arrangements
An even arrangement of the two barriers is characterized by
-matrices as given by
Equation 10. Substituting these matrices into
Equations 23,
24,
26, we obtain the conditions below for critical, supercritical, and bound states.
Critical states: Equation 23 simplifies to . This implies
Supercritical states: Equation 24 gives , which, in turn, implies
Bound states and resonances: Equation 26 now readsReal energy solutions for this equation give the bound-state energies, and the real part of the complex energy solutions gives the resonant energies.
We observe that the phase in Equation 10 does not have any effect on the spectrum of states that we are considering.
4.2.2 Odd arrangements
An odd arrangement of the two barriers is characterized by
-matrices as given by
Equation 15. By using
Equations 23,
24,
26with these matrices, we obtain the conditions below for critical, supercritical, and bound states.
Critical states: Equation 23 gives , which implies that
Supercritical states: Equation 24 gives , which gives
Bound states and resonances: Equation 26 now reads
For bound states, the left-hand side (lhs) must be real, and this condition will have solutions if and only if
We observe that single odd interactions necessarily have (or, in terms of the physical strengths, ; see Supplementary Appendix A or [45]). It is immediately clear from Equation 34 that an odd arrangement of two single odd-point interactions does not possess bound states.
Resonant energies will be given by the real part of the complex energy solutions of Equation 33.
Finally, as in even arrangements, the phase in Equation 15 is irrelevant to calculate the critical, supercritical, bound, and resonant energies for odd arrangements.
In the next section, we will consider some particular cases of even and odd arrangements of two-point interactions.
5 Some particular cases
In this section, we will consider some specific even and odd arrangements of a pair of point interactions. The cases we will consider are: i) equal mixtures of scalar and electrostatic strengths (which in the non-relativistic limit gives a pair of “ interactions”); ii) inverted mixtures of scalar and electrostatic interactions (resp. A pair of non-local “ interactions”); iii) a pair of pseudoscalar interactions (two “local interactions”); iv) a pair of magnetic interactions.
5.1 Even arrangements
5.1.1 Equal mixtures of scalar and electrostatic point interactions
We consider the even arrangement (
Equation 10) with each point interaction an equal mixture of electrostatic and scalar point interactions, that is,
arbitrary and
. It follows from
Supplementary Appendix Athat, in terms of the
-parameters, this corresponds to
,
, and
. In the non-relativistic limit, this equals two identical
interactions (for a brief review of the non-relativistic limit for all interactions considered in this work,
Supplementary Appendix B). Let us now investigate whether this double barrier admits critical, supercritical, or bound states.
Critical and supercritical states. Substituting the choice of parameters corresponding to an even, two-delta configuration in Equation 28, we find critical states for
The first case corresponds to the trivial case of a free particle at rest. On the other hand, because , Equation 29 for supercritical states is satisfied for any value of , and there is no absorption or emission of a bound state at the supercritical energy .
Bound states. In this case, the conditions for bound states, Equation 30, are reduced toor, after a slight manipulation,where we have made the replacement in the last line. It is clear that bound states can exist only for . In the equations above, the solution with the plus sign will be denoted by , and it stands for the ground state; accordingly, denotes the excited state.
Figure 1 shows a graph for the bound states. When varies from positive to negative values, a bound state (the ground state ) is absorbed from the continuum as crosses the boundary . Furthermore, this bound state is maintained (with decreasing energy) for every negative value of . When varies from toward , another bound state (the excited state ) is captured at the critical energy when . This bound state is also maintained as . By varying the parameter , there is no absorption (emission) of a bound state from (to) the continuum at the supercritical energy ; as we have seen above, the supercritical state is admitted for every value of .
It follows from Equation 35 that excited states exist only for a separation between the point interaction larger than the critical . In the limit , the energies become degenerate and tend to the bound state energy of a single interaction, . In Figure 1, both curves would coalesce into a single curve, with both the ground and the excited states captured as critical states at . On the other hand, in the limit , the capture of the excited state occurs at , and there is one bound state for a single point interaction with twice the strength (i.e., ) at , in agreement with Equation 21.
Resonances. To find the complex energy solutions for Equation 35, it is convenient to write it in the form (here we replace , for purely outgoing scattering solutions)
Now the lhs of this equation may be complex because and . The imaginary part of the lhs of this complex equation must be zero because (that is, and ) is real. Moreover, the imaginary part of this equation does not depend on the strength (equivalently, or ). Figure 2 shows a plot of the complex energies that solve Equation 366. We observe that in the limit , the double barrier becomes impermeable, and the complex energies that solve the equation become all real; these energies correspond to the infinite discrete set of energies allowed for the (or antiparticle) confined in the region . As the interaction strength increases in absolute value, the complex energies move alongside the “U” shaped blue and orange curves toward this discrete set of real energies. For and increasing in absolute value, the system captures a first bound state at , and at ( in the plot), the system captures the second bound state. As , both bound states approach the energy . For any value of the strength , there is a supercritical state at , which means that no bound state is emitted or absorbed at .
FIGURE 1
FIGURE 2
5.1.2 Inverted mixtures of scalar and electrostatic interactions
In this case, we choose
,
, which corresponds to
and
in
Equation 10. In the non-relativistic limit, each of the two-point interactions corresponds to a non-local
interaction.
Critical and supercritical states. Because in this case, , Equation 28 is trivially satisfied for any value of the parameter . Therefore, the system admits a critical state for every value of , and no bound state is emitted or absorbed at critical energy, , by varying . On the other hand, Equation 29 for supercritical states givesThe case corresponds to the trivial case of a free (anti-)particle at rest.
Bound states. In this case, after some manipulation, Equation 30 results inwhere in the last step, we have replaced . Note that this equation may be obtained from Equation 35 by making the substitution ; in fact, the same substitution explains the critical and supercritical states. Therefore, all the results for bound states and resonances for the present case follow from the results for an even arrangement of two-point interactions, each an equal mixture of electrostatic and scalar interactions simply by making , and the even arrangement of two inverted mixtures of scalar and electrostatic interactions behaves as an even arrangement of two equal mixtures of the same interactions acting on the “antiparticle sector.”
5.1.3 Two pseudoscalar interactions
The even arrangement of pseudoscalar interactions is obtained with the choicein Equations 6–9, which, due to Supplementary Appendix A6–A9, we have:
In the non-relativistic limit, each of the two-point interactions corresponds to the “local”
interaction. In the relationships above, both
cause
or
to diverge. These cases correspond to a pair of
impermeablepoint interactions (see
Equation 37in [
45]). The boundary conditions at the points
and
, in this case, depend on the value of
being
or
. If
, we have
and
. If, on the other hand,
, we have
and
.
Critical and supercritical states. Equations 28, 29 for the existence of critical and supercritical states, respectively, are always satisfied for any value of because . This means that, by varying (or, equivalently, by varying ), there is no absorption (emission) from (to) the continuum to (from) bound states.
Bound states. Equation 30 can now be written aswhere in the second line, we have taken into account that the lhs is strictly positive, and thus to find bound states, we must choose the plus sign when and the minus sign when . On the other hand, because , the lhs of the equation above is strictly greater than unity, whereas . Therefore, the equation above has no solution, and no bound states exist. This is consistent with the fact that a single pseudoscalar point interaction does not admit bound states because in the limits or , the even arrangement would have the same structure for bound states as a single point interaction.
Resonances. For complex energies, Equation 38, with , may have solutions for both signs and can be written more conveniently as
This equation has the symmetries and . As a result, it is sufficient to investigate its complex energy solutions for . Figure 3 shows the behavior of the complex energy solutions of Equation 39. The blue curve corresponds to the solution of the imaginary part of the equation, which does not depend on the strength . The colored points are the complex energies that solve both the real and complex parts of the complex equation for some particular values of , shown in the legend. As increases within this interval, the complex energies move toward the real axis alongside the blue curves. When , the double barrier becomes impermeable, and the set of real solutions corresponds to the allowed energies for a particle (or antiparticle) confined within the impenetrable walls at and .
FIGURE 3
5.1.4 Two magnetostatic interactions (two singular gauge fields)
In this case, we should consider the following parameters in Equations 6–9:which, due to Supplementary Appendix A6–A9, imply
That is, the only non-vanishing parameter in Equation 10 is the phase parameter . As mentioned in the previous section, this phase does not affect the existence of critical or supercritical states, nor the energies of bound states and resonances. Therefore, in what concerns these features, the system behaves as a free particle in this case.
5.1.5 Two scalar interactions
Here, we choose
arbitrary,
in
Equations 6–
9. Thus, from
Supplementary Appendix A6–A9, we have
For
, each one of the point barriers is
impermeable.
Critical and supercritical states. Equations 28, 29 give the same conditions for both critical and supercritical states, which areThe first of the conditions above corresponds to the free case, while the second one giveswhich have real solutions if, and only if, . The two solutions in Equation 41 are related by the transformation , which is a symmetry of the second of Equation 40. Thus, each value of satisfying the conditions above allows for both a critical and a supercritical state. As we will see below, at all values of satisfying Equation 40, there are simultaneous absorption and emission of a bound state at and .
Bound states. Equation 30 for this case can be written aswhich also has the symmetry and . Thus, to investigate the structure of the bound states and resonances, it is enough to consider .
Figure 4 shows the relationship between and the bound states. In this figure, the blue and the orange curves (indicated by and ) correspond respectively to the plus and minus signs in Equation 42. For any negative value of , the orange curve gives two bound states. When is between the roots of Equation 41, there are two additional bound states, given by the blue curves. There are two critical and two supercritical states corresponding to the values of satisfying Equation 41. There are also one supercritical and one critical state when , which correspond to a free antiparticle and a free particle at rest, respectively. When crosses 0 from above, two bound states (orange curve) are absorbed from the continuum at . By decreasing from zero toward negative values, two new bound states are absorbed from the continuum at when takes the value (blue curve). These states are later emitted into the continuum when takes the value . For , the two-point barriers become impermeable, and the system admits two nonzero energies in the interval (blue curve) for a particle confined between the two barriers (in which case the spinor vanishes identically outside ), as well as a state with energy (orange curve), with the spinor vanishing between the barriers and being nonzero outside them.7 If , only the two bound states given by the orange curve exist for any value of . In the limit , the orange curve yields the bound states of a single, scalar, point interaction with strength . Finally, for , the two-point interactions become independent, and the orange and blue curves coalesce to a single curve giving the bound states of a single scalar point interaction with strength .
Resonances. Equation 42, with , can be suitably rewritten asFigure 5 shows the complex energy solutions for this equation. The solid curves correspond to solutions of the imaginary part of that equation, which does not depend on the interaction strength . The colored points correspond to the solutions for the whole complex equation for several values of the strength . For , the two barriers become impenetrable, and the complex energies become real, with the set of discrete real energies for the cases and differing because the boundary conditions at and are different in the two cases. This set of real energies corresponds to the allowed energies for a particle (or antiparticle) confined between the two barriers. With within the interval but decreasing in absolute value, the complex energies move along the “U” shaped curves away from the real axis. There is the absorption/emission of two bound states when and another two when ( in the plot). The same complex solutions solve Equation 43 with .
FIGURE 4
FIGURE 5
5.1.6 Two electrostatic interactions
As the last special case of even arrangements, we will consider two electrostatic point interactions, which are obtained by assuming
arbitrary and
in
Equations 6–
9. Using
Supplementary Appendix A6–A9, this corresponds to take
in Equation 14. For any finite or infinite value of the strength
, each electrostatic point interaction is always permeable; see (
Supplementary Appendix A10). This has been interpreted as a signature of the Klein effect for point interactions because it implies that it is impossible to confine a particle (or antiparticle) on one side of a barrier by using only electrostatic interactions [
38].
Critical and supercritical states. Equation 28 for the existence of critical states now reads, in terms of the strength , as8The first of these equations corresponds to the free case, and the second one has the solutions
The solutions above satisfy , which is a symmetry of the second condition in Equation 44. Thus, for any value of the strength that admits a critical state, there will be a corresponding strength with inverted sign, , that also admits a critical state.
Similarly, Equation 29 for the existence of supercritical states gives
Here, too, we have the symmetry .
Bound states. Equation 30 can now be written aswhich, as expected, is also invariant under the transformation. Figure 6 shows the relationship between bound-state energies and the strength constant for an even arrangement of two identical electrostatic interactions, with the parameters and . The blue curve, denoted by , and the orange one, denoted by , are the solutions corresponding, respectively, to the plus and minus signs in Equation 45. We observe that there is at least one bound state for any value of , and it follows from the symmetry that for any bound state associated to the value , there corresponds a bound state, of same energy, associated to . As a consequence, the blue curves in the figure can be obtained from the orange ones through the transformation , and vice versa.
The critical (supercritical) states are admitted when the strength takes the values 0, , and . When the strength crosses from below, a bound state is absorbed from the continuum at the energy and, continuing to increase , a new bound state is absorbed from the continuum when at , becoming the new ground state. As continues to increase, the excited state (orange curve) is emitted into the continuum at when . When , there is always a single bound state. A similar behavior occurs when becomes more negative: a bound state is absorbed at when , another bound state occurs at when , and a bound state is emitted into the continuum at when ; when , the system admits a single bound state. When , the blue and the orange curves coalesce, corresponding to a single electrostatic point interaction with strength , as can be seen from Equation 45. On the other hand, when , the structure of the bound states is the same as that for a single electrostatic point interaction with strength , which always admit a single bound state (in this case, the bound states are emitted/absorbed when crosses , and the emission of a bound stated at is always simultaneous to the absorption of another one at , and vice versa).
Resonances. Now we seek complex energy solutions of Equation 45, with . From the symmetry , it would be enough to consider to investigate the structure of the resonances. Figure 7 shows the complex energy solutions of Equation 45. The blue and orange curves represent the solutions of the imaginary part of the equation for the plus and minus signs, respectively, and the colored points indicate the values of the complex energy that satisfy the full complex equation for selected values of the interaction strength , shown in the legend. As the interaction strength varies, the solution to the full equation travels along the blue and orange curves. In particular, at a specific value of the interaction strength ( in the plot), the solution reaches the supercritical energy , which corresponds to the absorption of a bound state. The bound state is eventually emitted at the critical energy (corresponding to the value in the plot).
FIGURE 6
FIGURE 7
5.2 Odd arrangements
5.2.1 Equal mixtures of scalar and electrostatic interactions
An odd arrangement of two equal mixtures of electrostatic and scalar interactions
corresponds to taking
,
and
in
Equation 15; see
Supplementary Appendix A. Below we analyze the critical, supercritical, bound, and resonant states, similarly to what we did for even arrangements.
Critical and supercritical states. Equation 31 implies for critical states, and Equation 32 yields a supercritical state for every value of . This means that there is no emission/absorption of bound states at the supercritical energy.
Bound states. The condition for bound states, Equation 33, may be rewritten aswhich is symmetric under the change . The system has a single bound state for any value of the strength , as shown in Figure 8; there is one particle bound state for any , and one antiparticle bound state for any . It follows from Equation 46 that in the limit , we obtain a free particle because the interactions cancel out. When , the bound state corresponds to that of a single equal mixture interaction; that is, .
It should be noted that our results for the bound states of both even and odd arrangements of equal mixtures of scalar and electrostatic interactions do not seem to coincide with those obtained in [66] for the same systems. It is likely that this discrepancy is due to the regularization and implicit prescriptions used in [66] to define the interactions. We note that our approach is rigorous and aligns with SAEs, where results from both approaches are available. In fact, the non-relativistic limit of Equation 37 coincides with the rigorously obtained results of [35].
Resonances. The complex energies that solve Equation 46 with are shown in Figure 9. The blue curve represents the solutions associated with the imaginary part of the equation, which is independent of the interaction strength . The discrete colored points correspond to solutions that satisfy the real and imaginary parts of Equation 46 simultaneously for a selection of values of . As the interaction strength varies, the complex solutions move in the complex energy plane along the curve shown. For , the solutions move toward the real axis as increases, while for , they shift horizontally. The spectrum is symmetric under the transformation . In the limit , the solutions become real and are given by the discrete energy levels of a particle or antiparticle confined between two impermeable walls located at and .
FIGURE 8
FIGURE 9
5.2.2 Inverted mixtures of scalar and electrostatic interactions
Here, we consider an odd arrangement of inverted mixtures of electrostatic and scalar interactions, obtained by assuming
,
in
Equations 11–
14. In terms of the
-parameters (see
Supplementary Appendix A), it follows that
,
, and
.
Critical and supercritical states. From Equations 31, 32, there is a critical state for every value of the strength because . On the other hand, a supercritical state exists only when . Therefore, there is no absorption/emission of bound states at a critical state.
Bound states and resonances. For this interaction, Equation 33 can be conveniently written in terms of (keeping in mind that ) aswhich can be promptly obtained from Equation 46 under the substitution (equivalently, ). Therefore, the entire structure of states for this interaction can be inferred from the results of the previous subsection. Recall that such a symmetry between the equal and inverted mixtures of electrostatic and scalar interactions was also observed in the case of even arrangements.
5.2.3 Two pseudoscalar interactions
Let us now consider an odd arrangement of two pseudoscalar point interactions, that is,
is arbitrary and
in
Equations 11–
14, which corresponds to
and
in Equation 20, according to the formulae in the
Supplementary Appendix A. A single pseudoscalar interaction is an odd interaction [
45]. Thus, this system is an odd arrangement of two single odd interactions and, as noted before [see comments after
Equation 34], this kind of system has no bound states.
Critical and supercritical states. Equations 31, 32 give critical and supercritical states for any value of the strength parameter . Therefore, there is no emission/absorption of bound states for this system.
Bound states. Equation 33 reduces toThe lhs of the above equation is a real number, whereas the rhs is purely imaginary because is real and . This equation has no solution for real , and thus, there is no bound state in this case.
Resonances. For the parameters above, with , Equation 33 can be written asThis equation is symmetric under and ; thus, it suffices to investigate the structure of resonances for . In particular, the case corresponds to two impenetrable walls at and . The imaginary part of Equation 47 is the same as for the even arrangement of pseudoscalar interactions (compare with Equation 39). Therefore, the even and odd arrangements will differ only by the solutions of the real part of the corresponding equations. Figure 10 shows the resonances for several values of the strength . The structure of the resonances is very similar to the one for the even arrangement of two pseudoscalar interactions; the difference is that now, the real parts of the complex energies are located on the complementary set of blue curves with respect to those in the even arrangement case. As a result, the real energies are approximately translated to the real axis.
FIGURE 10
5.2.4 Two magnetostatic point interactions
Similarly to the case of even arrangements, for odd arrangements, the two magnetostatic interactions also do not possess any structure of bound states, critical/supercritical states, or resonances because only the phase is non-vanishing in Equation 20. Thus, the system behaves essentially as a free particle.
5.2.5 Two scalar point interactions
In this case, the following parameters characterize the odd arrangement Equation 15: arbitrary, . From the expressions in Supplementary Appendix A, the -matrix parameters become , , and it follows that for , the two-point barriers are impenetrable.
Note that the transformation
has the effect of multiplying all the parameters
by
, which corresponds to multiplying the
-matrices in
Equation 15by a phase factor. Naturally, such a phase factor does not affect the conditions for the existence of physical states. Thus, it suffices to investigate the strength parameter
in the interval
.
Critical and supercritical states. Equations 31, 32 for the existence of critical and supercritical states, respectively, can only be satisfied for .
Bound states. In this case, Equation 33 may be written aswhich shows the additional symmetry . Figure 11 shows the relationships between the bound-state energies and the interaction strength . We note that for , there are always two bound states, with symmetric energies, for any value of the strength . For , these bound-state energies coalesce to zero energy, which is one of the allowed energies for a particle (or antiparticle) confined between the two impermeable walls at and . For , there are a critical and a supercritical state, as seen above.
Resonances. The complex energy solutions of Equation 48, with , are shown in Figure 12 for some values of the strength parameter . Note that Equation 48 implies the symmetry . Figure 12 shows that there are neither emissions nor absorptions of bound states at the critical and supercritical energies when . From the symmetry , we observe that as the interaction strength crosses the zero value, the bound-state energies are “reflected” at . As increases in the interval [0,2], the complex energies approach the real axis; for , the two-point barriers become impenetrable, and the energies form the set of discrete, real energies allowed for a particle (or antiparticle) confined between the two impenetrable walls at and . Similar conclusions for other values of can be drawn from the symmetries and .
FIGURE 11
FIGURE 12
5.2.6 Two electrostatic point interactions
As the final case, we now consider an odd arrangement of two electrostatic point interactions, that is,
arbitrary and
. It follows from
Supplementary Appendix Athat this choice of parameters corresponds to
and
in
Equation 15. As already mentioned when we considered the even arrangements in Subsection V A, the two single electrostatic barriers are never impenetrable because the parameters
are all finite (in terms of the interaction strengths, the permeability condition (
Supplementary Appendix A10) is satisfied).
Critical and supercritical states. Both Equation 28 and Equation 31 give the condition , which is equivalent to , for the existence of both critical and supercritical states.
Bound states. Substituting the above parameters in Equation 33, the condition for bound states becomes
This equation is symmetric under each of the transformations and . Figure 13 shows the relationship between the bound-state energies and the interaction strength . We observe that there are always two bound states, independently of the value of . We also observe that the energies of these two bound states have the symmetry , as is evident from Equation 49. This figure also shows that as crosses the zero value, there is neither emission nor absorption of bound states to or from the continuum, but only a “reflection” of the bound states at .
Resonances. Now we seek the complex energy solutions of Equation 49, with . Similarly to the previous cases, only the real part of that equation depends on the interaction strength . By taking into account the symmetries and , it is sufficient to investigate the resonances with the interaction strength in the interval . Figure 14 shows the structure of the resonances for three values of . As a consequence of the fact that electrostatic barriers are never impenetrable and thus are unable to confine a particle (or antiparticle) in the region between the point barriers, the resonances can never be real, regardless of the value of . As increases over the interval [0,2] the complex energy solutions (colored points in Figure 14) move along the blue curve toward the real axis but never reach it because when crosses the value , the colored points in the figure revert their movement along the blue curve. The behavior of the resonances for outside the interval [0,2] can be inferred from the symmetries and . Finally, the bound states are also represented in Figure 14 as colored points within the real interval . As increases within the interval , the bound-state energies move toward the origin, but never reach it; as crosses the value 2, the colored points reverse their motion within the bound-state interval .
FIGURE 13
FIGURE 14
5.3 Results summary
To end this section, we compile the results for the particular arrangements studied in two tables. Table 1 summarizes the results obtained for even arrangements, and Table 2 does the same for the odd ones.
TABLE 1
| Interaction type | Parameters | Critical states | Supercritical states | Bound states | Resonances |
|---|---|---|---|---|---|
| Scalar + electrostatic: two equal mixtures | arbitrary, | or ; bound state absorbed/emitted at these values | For all ; no bound-state absorption/emission | Only for . As decreases, the ground state appears at ; excited state appears at | Yes; become real at (impenetrable barriers) |
| Scalar + electrostatic: two inverted mixtures | arbitrary, | For all ; no bound-state absorption/emission | or ; bound states absorbed/emitted at these values | Same as two equal mixtures under | Same as two equal mixtures under |
| Two pseudoscalar | arbitrary, | For all ; no bound-state absorption/emission | For all ; no bound-state absorption/emission | None | Yes; become real at (impenetrable barriers) |
| Two magnetostatic | arbitrary, | None (free-like) | None (free-like) | None (free-like) | None (free-like) |
| Two scalar | arbitrary, | or ; real iff ; bound state absorption/emission at these values | or ; real iff ; bound state absorption/emission at these values | At least two for ; additional pair if when | Yes; become real at (impenetrable barriers) |
| Two electrostatic | arbitrary; | or with ; bound state absorption/emission at these values | or with ; bound state absorption/emission at these values | At least one for any ; number changes as crosses critical/supercritical values | Yes; never become real (barriers are always penetrable) |
Existence (or not) of resonances and of critical, supercritical, and bound states for even arrangements for all cases studied in the text.
TABLE 2
| Interaction type | Parameters | Critical states | Supercritical states | Bound states | Resonances |
|---|---|---|---|---|---|
| Scalar + electrostatic: two equal mixtures | arbitrary, | Only for | For all ; no bound-state absorption/emission | Exactly one for any : particle bound state for , antiparticle for | Yes; become real as (impenetrable barriers) |
| Scalar + electrostatic: two inverted mixtures | arbitrary, | For all ; no emission/absorption | Only for | Same as two equal mixtures under | Same as two equal mixtures under |
| Two pseudoscalar | arbitrary, | For all : no bound-state absorption/emission | For all ; no bound-state absorption/emission | None | Yes; become real at (impenetrable barriers) |
| Two magnetostatic | arbitrary, | None (free-like) | None (free-like) | None (free-like) | None (free-like) |
| Two scalar | arbitrary, | Only for | Only for | Two bound states for any with symmetry; for they coalesce at | Yes; become real at (impenetrable barriers) |
| Two electrostatic | arbitrary; | Only for | Only for | Always two bound states with symmetry | Yes; never become real (barriers are always penetrable) |
Existence (or not) of resonances and of critical, supercritical, and bound states for odd arrangements of interaction for all cases studied in the text.
6 Concluding remarks
We have investigated a relativistic, one-dimensional model with two contact interactions located symmetrically with respect to the origin, using the mathematically rigorous distributional method developed by some of the authors. Taking advantage of this approach, each of the contact interactions depends on four independent parameters with a well-defined physical meaning: scalar, electrostatic, magnetic, and pseudoscalar interactions. These interactions can be equivalently characterized by some matching conditions for the wave function at the points supporting the interaction. We discuss the form of the coefficients and matching conditions for two particular cases of special interest: even and odd interactions. We have also considered the limit of one barrier, in which both barriers are supported at the same point, for both even and odd barriers. Some interesting results have emerged after this limit: while the one-point limit of an even arrangement is always an even interaction, the limit of an odd arrangement does not always have a well-defined symmetry.
Using the form of the wave function outside the interaction points and the matching conditions at these points, we constructed the transmission matrix and, hence, the scattering matrix. Then, we provided a detailed analysis of the structure of critical, supercritical, and bound states, as well as of the resonances for even and odd arrangements of interactions for all the main relativistic, point interactions of physical interest in the literature, namely, equal and inverted mixtures of electrostatic and scalar interactions, pseudoscalar, scalar, magnetostatic, and electrostatic interactions.
A possible extension of this work would be to consider interactions with contact interactions, that is, interactions supported at points, and investigate the transmission bands. This requires the construction of the transfer matrix for the system of interactions. For an even number of interactions, the results obtained here can be directly applied because the comb can be seen as cells of two interactions. This will provide reflection and transmission coefficients and, hence, bound states and scattering features such as resonances. Additionally, as was demonstrated, in order to confine a particle or an antiparticle on one side of the interaction (i.e., to have an impermeable interaction), the single-point interactions must include a scalar and/or a pseudoscalar component because electrostatic interactions are non-confining (see Figures 7, 14). This could be related to the Klein effect and deserves further study.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
CB: Conceptualization, Formal Analysis, Investigation, Methodology, Supervision, Validation, Writing – original draft, Writing – review and editing. MG: Conceptualization, Formal Analysis, Funding acquisition, Investigation, Methodology, Supervision, Validation, Writing – original draft, Writing – review and editing. JL: Conceptualization, Formal Analysis, Funding acquisition, Investigation, Methodology, Supervision, Validation, Writing – original draft, Writing – review and editing. LM: Conceptualization, Formal Analysis, Funding acquisition, Investigation, Methodology, Supervision, Validation, Writing – original draft, Writing – review and editing.
Funding
The author(s) declared that financial support was received for this work and/or its publication. LM thanks the NASA—MN Space Grant Consortium and Concordia College’s Office of Undergraduate Research, Scholarship, and Creative Activity for partial financial support. This research was partially supported by the Q-CAYLE project, funded by the European Union-NextGenerationEU/MICIU/Plan de Recuperación, Transformación y Resiliencia/Junta de Castilla y León (PRTRC17.11), and also by projects PID2023-148409NB-I00, funded by MICIU/AEI/10. 13039/501100011033I. The authors also acknowledge the financial support of the Castilla y León Department of Education and the FEDER Funds (CLU-2023-1-05).
Acknowledgments
CB, JL, and LM thank the Departamento de Física Teórica, Atómica Y Óptica at Universidad de Valladolid, and in particular Professors Luis M. Nieto and Manuel Gadella, for the warm hospitality during visits at various stages of this work. MG thanks the Department of Mathematics and Statistics at the State University of Ponta Grossa and José T. Lunardi for the warm hospitality during a visit.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author JL declared that they were an editorial board member of Frontiers at the time of submission. This had no impact on the peer review process and the final decision.
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Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2026.1776432/full#supplementary-material
Footnotes
1.^This means the following: if is an arbitrary Lorentz vector, is a unit timelike vector, and is a component of the 2-dimensional Levi-Civita symbol, then transforms between compatible inertial frames as a Lorentz scalar field, as a component of a Lorentz vector field, and as a Lorentz pseudoscalar field, as shown by some of us for the case in [45].
2.^Notice that, according to the distributional approach, outside the support of the interaction, the distributional Dirac equation coincides with the usual Dirac equation.
3.^For the non-relativistic interpretation of the relativistic point interactions, it is often also convenient to rewrite the spinor in Equation 17 in the alternative form . See Supplementary Appendix B.
4.^In this case, the single point limit is degenerate to the free case, or the matrix is simply minus the identity matrix.
5.^The inverse of the imaginary part gives the mean life of the resonance [65].
6.^As observed in the introduction, the energies characterizing the resonances always come in complex pairs. However, in Figure 2 and all subsequent resonance figures, we only observe half of these pairs because we find the poles of the -matrix only for the principal component root, that is, for . Therefore, the other half of the resonances is trivially obtained by taking the complex conjugate energies.
7.^The state with in this case is analogous to the single bound state admissible for a single, scalar, impermeable barrier because the double impermeable barrier “cuts off” the region from the real line.
8.^If , we must have , and Equation 28 would imply , which is impossible to satisfy, given that .
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Summary
Keywords
bound-state energies, confining properties, one-dimensional Dirac equation, parity-symmetric interactions, point interactions, resonances, Schwartz distributions
Citation
Bonin CA, Gadella M, Lunardi JT and Manzoni LA (2026) Bound states and resonance analysis of one-dimensional relativistic parity-symmetric two-point interactions. Front. Phys. 14:1776432. doi: 10.3389/fphy.2026.1776432
Received
27 December 2025
Revised
08 February 2026
Accepted
13 February 2026
Published
01 May 2026
Volume
14 - 2026
Edited by
B. C. Chanyal, G. B. Pant University of Agriculture and Technology, India
Reviewed by
Rajesh Yadav, Sido Kanhu Murmu University, India
Fanfei Meng, China Agricultural University, China
Updates
Copyright
© 2026 Bonin, Gadella, Lunardi and Manzoni.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: José T. Lunardi, jttlunardi@uepg.br
ORCID: Carlos A. Bonin, orcid.org/0000-0002-9005-3100; Manuel Gadella, orcid.org/0000-0001-8860-990X; José T. Lunardi, orcid.org/0000-0001-7058-9592; Luiz A. Manzoni, orcid.org/0000-0002-0035-9529
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.