The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates

We investigate $L^1\to L^\infty$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\frac12}$ decay rate, which may be improved to $t^{-\frac12-\gamma}$ for any $0\leq \gamma<\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate except for a finite-rank piece. While in the case of a threshold eigenvalue only, the natural decay rate is preserved. In both cases we show that the decay rate may be improved at the cost of spatial weights.

We consider the massless case, when m = 0. For concreteness, we use There is much interest in the massless case due to its connection to graphene, see [24] for example. The Dirac equation was derived by Dirac as an attempt to connect the theories of quantum mechanics and special relativity. Dirac's derivation allowed for a model that is first order in time, as required for quantum mechanical interpretations while having a finite speed of propagation and allowing for external fields in a relativistically invariant manner. For a broader introduction to the Dirac equation, we refer the reader to the excellent text of Thaller, [31].
Much of the analysis in this paper will be based on properties of R 0 (λ) as λ → 0. It should be emphasized that while the Dirac and Schrödinger resolvents are closely related by (6), the massless Dirac operator has very different behavior from the massive Dirac or Schrödinger operators in the low energy regime. For example, R 0 (0) exists as a welldefined operator while R 0 (λ 2 ) has a logarithmic singularity at the origin and the resolvent of a massive Dirac operator has a logarithmic singularity at the threshold λ = ±m. These differences carry over into the low-energy asymptotic structure of resolvents of D 0 + V (x), which is again distinct from the threshold expansions for either Schrödinger or massive Dirac operators.
Detailed asymptotic expansions for the resolvents of both D 0 and its perturbations are computed in Section 3. For certain choices of potential, the operator D 0 + V (x) has an eigenvalue at zero. It is also possible for zero to be a non-regular point of the spectrum without an eigenvalue present, a phenomenon known as a resonance. We classify zero 1 Here and throughout the paper, scalar operators such as −∆ + m 2 − λ 2 are understood as (−∆ + m 2 − λ 2 )½ C 2 . Similarly, we denote L p (R 2 ) × L p (R 2 ) as L p (R 2 ). energy resonances and eigenvalues in terms of distributional solutions to Hψ = 0 in Section 7. We say that zero energy is regular if there are no distributional solutions to Hψ = 0 with ψ ∈ L ∞ (R 2 ), which may also be characterized by the uniform boundedness of the perturbed resolvent (D 0 + V − λ) −1 as λ → 0. We show that the classification of resonances for the massless Dirac equation and their dynamical consequences do not follow the same patterns as the Schrödinger equation.
iii) If there is only an eigenvalue at zero, then F t = 0.
We emphasize that our main results are the low energy bounds presented above. For the sake of completeness, we include the high energy result stated below.
We note that the added assumption on the lack of embedded eigenvalues is not needed for our low energy results in Theorem 1.1. The lack of embedded of eigenvalues has been established in the massive case, [8], and in the massless case for a sufficiently small potential, [11].
We establish the dispersive bounds by employing the functional calculus for the Dirac operator. For the class of potentials we consider, H is self-adjoint and the spectrum of H coincides with the real line. Under these circumstances, see [29], the Stone's formula for spectral measures yields: Here the perturbed resolvents are R ± V (λ) = lim ǫ→0 + (D 0 + V − (λ ± iǫ)) −1 , and their difference provides the spectral measure. We take advantage of the identity (6) to develop the spectral measure from Schrödinger resolvents. The Schrödinger free resolvent R ± 0 (λ 2 ) = lim ǫ→0 + (−∆ − (λ 2 ± iǫ)) −1 and the perturbed Schrödinger resolvent operators ) −1 are well-defined as operators between weighted L 2 (R 2 ) spaces, see [2].
To the authors' knowledge, this is the first study of dispersive estimates for the two dimensional massless Dirac equation. A recent paper of Cacciafesta and Seré, [10] investigated local smoothing estimates for the massless Dirac equation in dimensions two and three. The massive Dirac has been studied by the first and third author, [19], with Toprak [20]. The three-dimensional massive Dirac equation is more studied going back to the work of Boussaid [7], and D'Ancona and Fanelli, [13]. The characterization of threshold obstructions and their effect on the dispersive bounds have recently been studied by the first and third author and Toprak, [21]. Much of the work has roots in the study of other dispersive equations, notably the Schrödinger [28,30,22,17,18,14,32] and wave [13,26,4] equations.
Our low energy results in Theorem 1.1 establish the natural time decay t − 1 2 for the Dirac evolution while assuming less decay of the potential than has been required in the massive case. The improvement comes from using a more delicate argument based on Lipschitz continuity of the spectral measure, rather than direct integration by parts in the Stone's formula. A similar argument was used in [18].
In addition, this is the first result in which all the slow time decay caused by a pwave resonance is controlled in a finite rank term. Previous works on the Schrödinger or wave equation, [28,17,26], did not observe this asymptotic structure. Even in the weighted L 2 setting, [28], finite rank leading order terms had an error whose decay was only logarithmically better. The method we develop for computing spectral measures here can recover an analogous result (finite rank leading order, with polynomial decay of the remainder) for the Schrödinger evolution as well.
There is also much interest in the study of non-linear Dirac equations. See [23,5,12,9] for example. There is a longer history in the study of spectral properties of Dirac operators. Limiting absorption principles for the Dirac operators have been studied in [33,25,15,11]. In particular, the recent work [15] of the authors applies in all dimensions n ≥ 2 for both massive and massless equations, while the recent work of Carey, et. al. [11] applies to massless equations. The lack of embedded eigenvalues, singular continuous spectrum and other spectral properties is well established, [6,25,3,11,8]. In particular, for the class of potentials we consider, the Weyl criterion implies that σ ac (H) = σ(D 0 ) = (−∞, ∞).
There are no embedded eigenvalues provided the potential is small, see Theorem 3.15 in [11].
The paper is organized as follows. We begin by proving the natural dispersive estimates for the free massless Dirac operation in Section 2. In Section 3 we develop a variety of expansions for the free resolvent that will be needed to study the spectral measure in (7).
In Section 4 we prove Theorem 1.1 when zero energy is regular. In Section 5 we establish more delicate expansions of the perturbed resolvent around the threshold in the presence of resonances and/or eigenvalues so that we may prove Theorem 1.1 when the threshold is not regular in Section 6. In Section 7 we provide a characterization of the threshold obstructions that relates them naturally to the various subspaces of L 2 that arise in the resolvent expansions. Finally, Section 8 contains the various integral estimates needed throughout the paper.

Free Dirac dispersive estimates
Due to the relationship between the massless free Dirac evolution and the free wave equation, D 2 0 f = −∆f , we can expect a natural time decay rate of size |t| − 1 2 as one has in the wave equation (when m = 0) provided the initial data has more than 3 2 weak derivatives in L 1 (R 2 ). In the case of Dirac equation, as in Schrödinger equation, the time decay can be improved at the cost of spatial weights.
Further, one has The proof of this theorem is based on asymptotic expansions of the spectral measure of the free Dirac operator, both at low energies and high energies. To best utilize these expansions, we employ the notation The notation primarily refers to derivatives with respect to the spectral variable λ in the expansions for the integral kernel of the free resolvent operator. In the context of (6), due to the gradient, we use the O(g) to refer to |x − y| as well. If the derivative bounds hold only for the first k derivatives we write f = O k (g). In addition, if we write f = O k (1), we mean that differentiation up to order k is comparable to division by λ and/or |x − y|.
This notation applies to operators as well as scalar functions; the meaning should be clear from the context.
We also state two other bounds for µ 0 which will be useful in later sections. The interpolation argument above also implies that (14) |µ Similarly, using (11) and (12) we obtain the bound Using the support of χ(λ) in the definition of µ 0 , it is easy to see that R e −itλ µ 0 (λ)(x, y) dλ 1.
For |t| 1, again using the support of χ(λ) and (13), we have For the weighted bounds, after two integration by parts, we have Interpolating these bounds we conclude for any γ ∈ [0, 3 2 For large energies, to prove the first claim it suffices to bound is comparable to λJ 0 (λ|x − y|), see (9) and [1]. Using Lemmas 3.2 and 5.3 in [26], we have the bounds However these estimates rely on oscillation that may not be present when t is small. To obtain a uniform bound for small times, the integrand must be absolutely convergent.
Given the growth of |ω ± (λ|x − y|)| |λ|, we need a multiplier that decays like uniformly in x and y for small t. The additional powers of λ correspond to extra mollifi- Similar ideas can be used to prove Theorem 1.2. One can essentially reduce to the high energy results for the wave equation derived in Propositions 3.1 and 5.2 of [26]. We leave the details to the interested reader. The extra smoothing powers of H, represented with negative powers of λ, are required since the perturbed Dirac resolvent doesn't decay in the spectral variable. One also needs the continuity of the potential to use the limiting absorption principle for the Dirac operator(s) proven by the authors in [15].

Free resolvent expansions around zero energy
In this section we study the behavior of the free Dirac resolvent more carefully by using the properties of free Schrödinger resolvent R 0 (λ) = (−∆−λ) −1 . Following [30,17,18,20], we have the following expansion for the Schrödinger resolvent.
As a corollary we have the following Lipschitz bounds. The 1 2 -Lipschitz bound cannot be improved without growth in |x − y|, which leads to weights in the dispersive bounds, due to the large λ|x − y| term.
In the case when zero is not regular, we will need a further expansion of R ± 0 : Lemma 3.4. We have the expansion for the kernel of the free resolvent Further, when |λ| ≤ 1, the error term satisfies Moreover, for 0 ≤ γ < 1 2 and |λ 1 | ≤ |λ 2 | 1, we have Proof. The first bound for the error term follows from (27) when |λ||x − y| ≪ 1. When |λ||x − y| 1, it follows by writing and for |λ||x − y| 1 we have Using these bounds with 1 2 ≤ k < 1, we obtain the Lipschitz bound by interpolating the trivial bound, with the bound we obtain using the mean value theorem:

Small energy dispersive estimates when zero is regular
As usual, see for example [30,17,19,21,20], we use the symmetric resolvent identity to understand the low energy evolution. In the Dirac context the potentials are matrixvalued, and we have the assumption that the matrix V : we may use the spectral theorem to write To employ the symmetric identity, with η j = |ζ j | 1 2 , we write Note that the entries of v are Define the operators and let Definition 4.1. We make the following definitions that characterize zero energy obstruc- ii) We say there is a resonance of the first kind at zero if T is not invertible on L 2 , but iii) We say there is a resonance of the second kind at zero if S 1 vG 1,1 v * S 1 is not invertible.
iv) Let S 2 be the Riesz projection onto the kernel of S 1 vG 1,1 v * S 1 , then S 1 − S 2 has rank at most two and S 1 − S 2 = 0 corresponds to the existence of 'p-wave' resonances at zero. S 2 = 0 corresponds to the existence of an eigenvalue at zero. In contrast to the massive case, see [19], there are no 's-wave' resonances in the massless case. See Section 7 below for a complete characterization.
v) Noting that vG 0,0 v * is compact and self-adjoint, T = U + vG 0,0 v * is a compact perturbation of U. Since the spectrum of U is in {±1}, zero is an isolated point of the spectrum of T and the kernel is finite dimensional. It then follows that S 1 is a finite rank projection, and since S 2 ≤ S 1 , so is S 2 .
We employ the following terminology from [30,17,18]: We say an operator T : We note that Hilbert-Schmidt and finite-rank operators are absolutely bounded operators.
We now concentrate on the case when zero is regular. The following expansions for M ± (λ) around zero energy suffice in this case.
In all statements above the error terms are understood in the Hilbert-Schmidt norm.
The following lemma establishes analogous bounds for (M ± (λ)) −1 when zero is regular.
x −β and that zero is a regular point of the spectrum.
In all statements above the error terms are understood as absolutely bounded operators.
Proof. When zero is regular, the operator T is invertible with an absolutely bounded inverse. Therefore, by Lemma 4.3, M ± (λ) is invertible with a uniformly bounded inverse provided that 0 < |λ| ≪ 1 and |V (x)| x −2− .
Using resolvent identity, the boundedness of (M ± ) −1 and (44) we obtain (47): To obtain (48), we use (45) and the identity Finally, (49) follows from (47), (45) and (48) after writing We are now ready to prove the small energy assertions of Theorem 1.1 when zero is regular by studying the small energy portion of the Stone's formula, (7), In particular, we will prove the following family of bounds, which includes the uniform bound when γ = 0.
Proposition 4.5. Fix 0 ≤ γ < 3 2 and assume that |V (x)| x −2−2γ− . If zero is regular, In [19], the authors studied the solution operator as an operator H 1 → BMO because the operator G 0,0 is not bounded from L 1 → L 2 or from L 2 → L ∞ . Simple use of iterated resolvent identity was not enough to deal with this problem in the massive case since one relies on the orthogonality properties of the most singular terms in the expansion of the operator M ± (λ) −1 = U + vR ± 0 (λ)v * −1 to get uniform estimates in x, y. In [20], this problem was overcome by selectively using the iterated resolvent identity for M ± (λ) −1 only for certain terms arising in the expansion.
Since we don't rely on orthogonality arguments here, we need only use the iterated symmetric resolvent identity: We consider the contribution of the first three summands in (51) to the Stone's formula.
Proof. The contribution of the first term is the free evolution which was dealt with above in Theorem 2.1. We note the following useful algebraic identity It suffices to consider the contribution of the following to the integral ). The remaining terms have similar structure with differences µ 0 on the right instead of the left.
Using the bounds (10) and (33), and noting Lemma 8.2, we see that the kernel of Γ is bounded in λ, x, y and it is supported in |λ| 1. Therefore, we restrict our attention to the case |t| > 1.
The lemma below takes care of the contribution of M −1 term for 0 ≤ γ < 1 2 . In contrast to the massive case [19,20] or Schrödinger [18], for the massless Dirac bound, the argument employed here does not require any cancellation between the '+' and '-' terms in the Stone's formula, (7).
be an absolutely bounded operator satisfying (for |λ|, |λ 1 |, |λ 2 | 1 with |λ 1 | ≤ |λ 2 |) Note that the hypothesis is satisfied by the mean value theorem if T (λ) = O 1 (λ − 1 2 + ) as an absolutely bounded operator. Also note that when zero is regular M −1 satisfies the Proof. Dropping ± signs, let R := vR 0 V R 0 . Using the support of χ(λ) as well as the bounds (33) and (37) for the free resolvent and the integral estimates in Lemmas 8.4 and Note that (54) and Lemma 8.2 imply that L 2 y 1 norm of R(λ)(y 1 , y) is bounded uniformly in y and λ, while (55) implies that the L 2 y 1 norm of R(λ 1 )(y 1 , y) − R(λ 2 )(y 1 , y) is bounded by y γ |λ 1 − λ 2 | 1 2 +γ . Using these bounds and the hypothesis for T , using (52) we see that (with Γ := We use (33) and (37) for the free resolvent terms. Therefore, by applying the Lipschitz argument as in (16) and the proof of Lemma 4.6, we bound the integral by For 1 2 ≤ γ < 3 2 , we have the following lemma which we state only for M −1 . We dropped ± signs since we won't rely on any cancellation between ± terms.
We now prove Proposition 4.5.
Proof of Proposition 4.5. Using the expansion (51), we see that the first terms are con- suffices to establish the desired bound for 0 ≤ γ < 1 2 . The case 1 2 ≤ γ < 3 2 is established in Lemma 4.8.

Small energy resolvent expansion when zero is not regular
We now consider the case when zero is not a regular point of the spectrum. We first provide the necessary expansions to develop the spectral measure when there are eigenvalues and/or resonances at zero energy, then establish the dispersive estimates. We re-emphasize here that this is the first result, to our knowledge, in which the contribution of a 'p-wave' resonance is controlled in a finite-rank term. Previous results in the Schrödinger (or wave equation) context, [28,17,26], have not achieved this. Even in the weighted L 2 setting, [28], any finite rank pieces had an error whose decay was only logarithmically better. This argument can be modified to apply to the Schrödinger evolution as well.
With S 1 being the Riesz projection onto the kernel of T , define (T + S 1 ) −1 := T 1 . One can see that S 1 T 1 = T 1 S 1 = S 1 . Then, we have the following variations of Lemma 4.3 and Lemma 4.4.
Proof. The first assertion follows from the invertibility of T + S 1 , (43) and a Neumann Series computation. Recalling that T 1 = (T + S 1 ) −1 , the expansion (58) follows from Lemma 5.1 noting that The proof of (59) is identical to the proof of (47). Finally (60) follows from the Lipschitz bound for E ± 1 in Lemma 5.1, the bound Γ = O(|λ| 1− ), and by noting that the first two terms in the definition of Γ satisfies the Lipschitz bound To invert M ± (λ) = U + vR ± 0 (λ 2 )v, for small λ, we use the following lemma (see Lemma 2.1 in [27]) repeatedly.
We apply this lemma with M = M ± (λ) and S = S 1 . The fact that M ± (λ) + S 1 has a bounded inverse in L 2 (R 2 ) follows from Lemma 5.2. We also need to prove that has a bounded inverse in S 1 L 2 (R 2 ). We have, using (58) and the fact that S 1 T 1 = S 1 , We write: The remainder of this section is devoted to inverting A ± (λ) in a neighborhood of zero under different spectral assumptions.

Proposition 5.4. Assume that |V (x)|
x −2− . For sufficiently small λ, the operators A ± (λ) are invertible on S 1 L 2 . Further, as an operator on S 1 L 2 . Morever which is independent of λ and the choice of sign.
We note that these operators are finite rank on L 2 since S 1 L 2 is a finite-dimensional subspace.
Proof. We begin by writing the projection We note that Q corresponds to a projection onto the p-wave resonance space, and by Corollary 7.3, has rank at most two. We first note that when Q = 0, the statement follows (63) and the orthogonality property that S 2 vG 1,1 = 0. The invertibility of the resulting operator is guaranteed by Lemma 7.6. The following lemma implies the proposition when S 2 = 0.
Lemma 5.5. When Q = 0, the operator QA ± (λ)Q is invertible for sufficiently small λ. Further, as an operator on QL 2 . Morever Proof. We begin by showing that QA ± (λ)Q is invertible on QL 2 . In the case that Q has rank one, then using (63) we can see that QA ± (λ)Q is a scalar of the form Which, by (20), suffices to show our desired results.
We now write with respect to the basis {φ 1 , φ 2 }: where A 1 is a 2 × 2 matrix of constants given by the contributions of φ i vG 1,0 v * φ j . Since G 1,1 v * φ 1 and G 1,1 v * φ 2 are linearly independent, the first matrix above is invertible, and hence, for sufficiently small λ, QA ± (λ)Q is invertible. Moreover the entries of its inverse are rational functions in log(λ), and the degree of the denominator is at least one more than the degree of the numerator. In particular, they are of the form O 1 ( 1 log(λ) ). The final claim follows from the resolvent identity and (20), since (A + − A − )(λ) is independent of λ.
We are now ready to obtain a suitable expansion for M ± (λ) −1 when zero is not regular. Note that Proposition 5.4 and its proof gives detailed expansions for A ± (λ) −1 , in particular, the projection Q corresponds to the contribution of p-wave resonances and the operator [S 2 vG 1,0 v * S 2 ] −1 to the threshold eigenspace, see Lemma 7.7 below.
Proof. Using Lemma 5.3 with M = M ± (λ) and S = S 1 , and recalling that Using Lemma 5.6, we have Since by Lemma 5.2 the operator (M ± (λ) + S 1 ) −1 satisfies better Lipschitz bounds than E ± 3 (λ), and since the last two terms are similar, we concentrate on the term By Lemma 58 we have (M ± (λ) + S 1 ) −1 = O(1), Combining this with (20) we see that The Lipschitz bound follows by using the bounds above and in addition the bounds in Lemma 5.2 for (M ± (λ) + S 1 ) −1 , and by noting that The contribution of E ± 3 is controlled by the bound in Lemma 5.6, specifically (65). For the contribution of the remaining terms, we note Then (66) suffices to control the second term, while the first term is controlled by using (A ± (λ)) −1 = O(1) by Proposition 5.4 and the simple bound 6. Small energy dispersive estimates when zero is not regular In this section we study the small energy portion of the Stone's formula, (7), when zero is not regular: In particular, we prove the following result.
In fact, when zero is not regular we explicitly construct the finite rank operator F t , see (69) below.
Proof of Proposition 6.1. Recall (51). As in the regular case, Lemma 4.6 suffices to control the first few terms arising in (51), hence we turn our attention to the tail. Recall that by Lemma 5.7 we have . The contribution of the second term in the Stone's formula is taken care of by Lemma 4.7 by taking k = 1 2 + γ+ in the error bounds for E ± 4 (λ). This requires that β > 3 + 2γ. It remains to consider the contribution of If we replace at least one of the free resolvents with R ± 0 − G 0,0 , we obtain further λ smallness which allows us to obtain the desired t − 1 2 bound with minor modifications of the proof of Lemma 4.7. In particular, we note that Further, Iterating this process, we may write We first consider the contribution of the first term to the Stone's formula. When there is a p-wave resonance at zero, when S 1 − S 2 = 0, using Proposition 5.4, the ± difference easily yields a finite rank term with logarithmic decay in time since So when there is a 'p-wave' resonance at zero, we can explicitly construct the operator In the eigenvalue only case, when S 1 = S 2 = 0, by Proposition 5.4 the leading term in (68) disappears by ± cancellation since A ± (λ) −1 is independent of the choice of sign in this case. Therefore F t = 0.
In particular, the L 2 y 1 norm of R(λ)(y 1 , y) is bounded in y and λ, and the L 2 (33), Therefore R satisfies the following improved pointwise bound In particular, the L 2 y 1 norm of R(λ)(y 1 , y) is bounded by |λ| 1− . Using these bounds and the hypothesis for T , we see that (with Γ := This implies the uniform bound when t is small. Also using this in the case |λ 1 −λ 2 | |λ 2 | we obtain The first summand above corresponds to the case when the difference is on T and the second summand corresponds to the remaining cases. Combining these bounds for 0 ≤ γ < 1 2 we have Therefore, by applying the Lipschitz argument as in (16), we bound the integral by This finishes the proof of Proposition 6.1.

Threshold characterization
The characterization of the threshold is similar to the characterization for the massive case in [19]. See [21] for the three dimensional threshold characterization. These results have roots in the characterizations for Schrödinger operators may be found in [22,17,14].
with ψ a distributional solution to Hψ = 0 and ψ ∈ L p (R 2 ) for all p > 2.
Proof. Take φ ∈ ker(T ), φ ∈ L 2 . Then Here, recalling (28) and (3), we have That is, if φ ∈ ker(T ) we have Hψ = 0. Now, to show that ψ ∈ L p , we have ψ = −G 0,0 v * φ with φ ∈ L 2 . We can bound |G 0,0 (x, y)| |x−y| −1 to employ a fractional integral operator argument. So that, Proof. By the last lemma, we have ψ ∈ L ∞ . We recall that ψ = −G 0,0 v * φ and the kernel of G 1,1 is 1, so The first term is in L 2 (see Lemma 7.3 in [19]). Combining this with ψ ∈ L ∞ finishes the proof. We note that the assumption that β > 2 suffices here, the logarithmic terms in the massive case considered in [19] required further decay of the potential. These terms do not occur in the massless case, specifically we need only (68) in [19] for which β > 2 is sufficient.
Corollary 7.3. The rank of S 1 is at most two plus the dimension of the eigenspace at zero.
We note that the at most two dimensional space of resonances correspond to the p-wave resonances in the massive Dirac, [19], and Schrödinger [17] operators. We again note that there are no 's-wave' resonances in the massless case.
Proof. By Lemma 7.2, ψ ∈ L 2 if and only if G 1,1 v * φ = 0, which is equivalent to φ being in the kernel of S 1 vG 1,1 v * S 1 .
We now prove that S 2 vG 1,0 v * S 2 is always invertible on S 2 L 2 .
Lemma 7.7. The projection onto the zero energy eigenspace is The proof follows along the lines of Lemma 7.10 in [19]. For the sake of brevity, we omit the proof.

Integral Estimates
Finally, we provide proof of the integral estimates that are used throughout the paper.
The final estimate is proven similarly.