Dispersive estimates for Schr\"odinger operators in dimension two with obstructions at zero energy

We investigate $L^1(\R^2)\to L^\infty(\R^2)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave resonance at zero energy does not destroy the $t^{-1}$ decay rate. We also show that if there is a p-wave resonance or an eigenvalue at zero energy then there is a time dependent operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim 1$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{-1}, \text{for} |t|>1.$$ We also establish a weighted dispersive estimate with $t^{-1}$ decay rate in the case when there is an eigenvalue at zero energy but no resonances.


Introduction
Consider the Schrödinger operator H = −∆ + V in R 2 , where V is a real-valued potential. Let P ac be the orthogonal projection onto the absolutely continuous subspace of L 2 (R 2 ), which is determined by H. In [23], Schlag proved that e itH P ac L 1 (R 2 )→L ∞ (R 2 ) |t| −1 under the decay assumption |V | x −3− and the assumption that zero is neither an eigenvalue nor a resonance of H.
Recall that (see, e.g., [14] or Section 5 below) there is a resonance at zero energy if there is a distributional solution to the equation Hψ = 0 where ψ / ∈ L 2 (R 2 ) but ψ ∈ L p (R 2 ) for some p ∈ (2, ∞]. There are two possible cases, either ψ ∈ L ∞ (R 2 ) and ψ / ∈ L p (R 2 ) for any p < ∞ or ψ ∈ L p (R 2 ) for all p ∈ (2, ∞]. In the case of ψ ∈ L ∞ (R 2 ) only, the resonance is called an s-wave resonance. In the second case, we say there is a p-wave resonance. We say that there is an eigenvalue at zero if ψ ∈ L 2 (R 2 ).
This definition for resonances differs from the case of dimension n = 3 in which ψ lies in weighted L 2 spaces.
We note that in the case of V ≡ 0 the function ψ ≡ 1 solves Hψ = 0 which corresponds to an s-wave resonance. It is important to note that in spite of this obstruction, the free evolution decays in time at the rate t −1 .
The first author was partially supported by NSF grant DMS-0900865. 1 Much is known about dispersive estimates for the Schrödinger equation when zero is regular. The history goes back to Rauch, [19], who studied the local decay in dimension three. In [19], he noted that in the generic case, i.e. when there may be eigenvalues or resonances, the evolution decays at a rate of |t| −1/2 as t → ∞ on exponentially weighted L 2 spaces. In the case when there are no eigenvalues or resonances, it was shown that the decay rate is |t| −3/2 . Jensen and Kato, [13], improved this result to polynomially weighted L 2 spaces in dimension three, and higher dimensions, [11,12]. In [13], it was noted that the presence of a zero energy eigenvalue or resonance destroys the |t| −3/2 decay even if one projects away from the eigenspace in dimension three.
Local decay estimates in the two dimensional case when zero is regular were studied by Murata in [18]. Murata was able to prove an estimate on weighted L 2 spaces that decays like t −1 (log t) −2 , which is integrable at infinity. Such estimates have been used in analysis of the stability of certain two-dimensional non-linear equations.
The first result to discuss global decay, L 1 → L ∞ estimates, was due to Journé, Soffer and Sogge in [16]. Their result relied on the integrability of t −n/2 at infinity and is thus restricted to n ≥ 3. Much is now known in this direction, mainly in dimension three. Rodnianski and Schlag established such estimates in dimension three, [21], in addition to establishing Strichartz estimates. Following from their methods, a great number of results in dimension three followed, particularly [8,9,10]. The one dimensional problem was studied by Weder, [25] and Goldberg and Schlag [10]. Also see [5,27,7] for global estimates in the three-dimensional case when there is an eigenvalue and/or resonance at zero energy, and [6] for a similar result for the matrix Schrödinger equation.
There have also been studies of the wave-operators in dimension two. In particular Yajima, [26] established that the wave operators are bounded on L p (R 2 ) for 1 < p < ∞ if zero is regular. The hypotheses on the potential V were relaxed slightly in [15]. This result would imply global dispersive estimates if extended to the full range of p, 1 ≤ p ≤ ∞. High frequency dispersive estimates, similar to those obtained in [23] stated as Theorem 1.3 below were obtained by Moulin, [17], under an integrability condition on the potential.
In this paper we investigate L 1 → L ∞ dispersive estimates in R 2 when zero energy is not a regular point of the spectrum of the operator H = −∆ + V . Our goal is to prove the following theorem. If there is a p-wave resonance or eigenvalue at zero, then for β > 6, there is a time-dependent operator F t such that Note that it is natural to have the t −1 decay rate in the case of an s-wave resonance since the free Schrödinger has an s-wave resonance at zero energy. The reason that we can not get any decay in the case of a p-wave resonance or the zero eigenvalue is the behavior of the resolvent around zero energy.
In the three dimensional case the resolvent (H − z 2 ) −1 has an expansion of the form The most singular term G −2 z −2 gives the Riesz projection to zero energy eigenspace. If one projects away from the zero eigenspace, the worst singularity is 1 z , which allows for |t| −1/2 decay as t → ±∞, see [5]. However, in the two dimensional case the resolvent expansion around zero contains logarithmic terms. In particular, in the general case of zero energy resonances (even if one projects away the zero energy eigenspace), the most singular term is of the form 1 z 2 log(z) , which does not allow for any polynomial decay in t. It may be possible to get a decay of the form 1 log(t) as in [18] but we won't pursue this issue here. However, it is possible to improve this theorem in the case when zero is an eigenvalue but there are no resonances at zero. In particular, we show the following. Here L 1,1+ is the weighted L 1 space defined by L 1,1+ (R 2 ) := {f : Let χ is an even smooth function supported in [−λ 1 , λ 1 ] and χ(x) = 1 for |x| < λ 1 /2. Let K λ1 be the kernel of e itH χ(H)P ac : is the perturbed resolvent. By the limiting absorption principle, these boundary values are bounded operators on weighted L 2 -spaces, see e.g. [2].
Therefore, in the proof of Theorem 1.1 and Theorem 1.2, it suffices to obtain the stated bounds for the operator K λ1 for some λ 1 > 0. Our analysis relies on expansions of the resolvent operator at zero energy following those of [14], also see the previous work in [3,4]. We repeat part of the argument to obtain more flexible and favorable error bounds for our purposes.
We also note that standard spectral theoretic results for H apply. Under our assumptions we have that the spectrum of H can be expressed as the absolutely continuous spectrum, the interval [0, ∞), and finitely many eigenvalues of finite multiplicity on (−∞, 0]. See [20] for spectral theory and [24] for Birman-Schwinger type bounds.
Our paper is organized as follows. We set out the necessary expansions for the resolvent in Section 2.
We then study K λ1 to establish Theorem 1.1 in the case when there is an s-wave resonance at zero in Section 3. In Section 4 we establish Theorem 1.1 in the case of a p-wave resonance or eigenvalue at zero energy. In Section 5 we discuss the spectral structure of −∆ + V at zero energy. Finally we prove Theorem 1.2 in Section 6.

Resolvent expansions around zero energy in the case of an s-wave resonance
In this section, following [14], we obtain resolvent expansions around the threshold λ = 0 in the case when there is only s-wave resonance at zero (resonance of the first kind, see Definition 2.3 below and the remarks following it). We now introduce some definitions and notation.
It is worth noting that a Hilbert-Schmidt operator is an absolutely bounded operator.
We say that an absolutely bounded operator T (λ)(·, ·) is O 1 (λ s ) if the integral kernel satisfies the following estimates: If only the first bound in (2) holds, we say that T (λ)(·, ·) is O(λ s ). We also note that we can replace Recall that where H ± 0 are the Hankel functions of order zero: From the series expansions for the Bessel functions, see [1], as z → 0 we have We also have the following estimates for the derivatives as z → 0 Further, for |z| > 1, we have the representation (see, e.g., [1]) This implies that for |z| > 1 We use the symmetric resolvent identity, valid for ℑλ > 0: The key issue in the resolvent expansions is the invertibility of the operator M ± (λ) for small λ under various spectral assumptions at zero. Below, we obtain expansions of the operator M ± (λ) around λ = 0 using the properties of the free resolvent listed above. A similar lemma was proved in [23], however we need to expand the operator further and obtain slightly more general error bounds. The following operators arise naturally in the expansion of M ± (λ) (see (4), (5)) Here g ± (λ) = a ln λ + z where a ∈ R\{0} and z ∈ C\R, and T = U + vG 0 v where G 0 is an integral operator defined in (11). Further, for any 1 2 ≤ k < 2, Here G 1 , G 2 are integral operators defined in (12), (13), and g ± 1 (λ) = λ 2 (α log λ+β ± ) where α ∈ R\{0} and β ± ∈ C\R. Further, for any 2 < ℓ < 4, Proof. The first part with k = 1 2 was proven in [23, Lemma 5]. To obtain the expansions recall that, for λ > 0, Using the definition of H ± 0 (z), and the expansions (4) and (5) around z = 0, we have with α = 1/8π and β ± ∈ C. The expansions are now obtained by setting z = λ|x − y|. In particular, we see Noting that Using (16) and (15) for M 0 and M 1 respectively, we obtain for z = λ|x − y| < 1 that For large z, using the expansion of the Hankel function about z = ∞, recall (8), we have |H ± 0 (z)| 1 and | d dz H ± 0 (z)| z −1/2 . So that for large z > 1, for M ± 0 (z) the log z term dominates and for M ± 1 (z) the z 2 log z term in (15) dominates, and we have Hence, for any 0 < k < 2, and for any 2 < ℓ < 4 we have This yields the claim for M 0 and for β > 1 + ℓ. For λ-derivatives, we note that and For the terms in M 0 and M 1 other than R 0 , the effect of ∂ λ is comparable to division by λ. However, due to (18), on λ|x − y| > 1 we have for any k ≥ 1 2 , Similarly, We now give the definition of resonances from [14], also see [23]. Recall that Q := ½ − P .

Definition 2.3.
(1) We say zero is a regular point of the spectrum of H = −∆ + V provided (2) Assume that zero is not a regular point of the spectrum. Let S 1 be the Riesz projection onto the kernel of QT Q as an operator on QL 2 (R 2 ). Then QT Q + S 1 is invertible on QL 2 (R 2 ).
Accordingly, we define D 0 = (QT Q + S 1 ) −1 as an operator on QL 2 (R 2 ). We say there is a resonance of the first kind at zero if the operator T 1 := S 1 T P T S 1 is invertible on S 1 L 2 (R 2 ).
(3) We say there is a resonance of the second kind at zero if T 1 is not invertible on S 1 L 2 (R 2 ) but , where S 2 is the Riesz projection onto the kernel of T 1 (recall the definition of G 1 and G 2 in (12) and (13)).
(4) Finally, if T 2 is not invertible on S 2 L 2 (R 2 ), we say there is a resonance of the third kind at zero. We note that in this case the operator where S 3 is the Riesz projection onto the kernel of T 2 (see (6.41) in [14] or Section 5 below).
Remarks. i) In [14], it is noted that the projections S 1 − S 2 , S 2 − S 3 and S 3 correspond to s-wave resonances, p-wave resonances, and zero eigenspace respectively. In particular, resonance of the first kind means that there is only an s-wave resonance at zero. Resonance of the second kind means that there is a p-wave resonance, and there may or may not be an s-wave resonance. Finally, resonance of the third kind means that zero is an eigenvalue, and there may or may not be s-wave and p-wave resonances. We characterize these projections in Section 5.
ii) Since QT Q is self-adjoint, S 1 is the orthogonal projection onto the kernel of QT Q, and we have This statement also valid for S 2 and (T 1 + S 2 ) −1 , and for S 3 and (T 2 + S 3 ) −1 .
iii) The operator QD 0 Q is absolutely bounded in L 2 . This was proved in Lemma 8 of [23] in the case S 1 = 0. With minor modifications, the same proof works in our case, too.
iv) The operators with kernel vG i v are Hilbert-Schmidt operators on if i = 1 and β > 3 for i = 2, 3.
We will apply this lemma with A = M ± (λ) and S = S 1 . Thus, we need to show that M ± (λ) + S 1 has a bounded inverse in L 2 (R 2 ) and has a bounded inverse in S 1 L 2 (R 2 ). We prove these claims and obtain expansions for the inverses for each type of resonance in Lemma 2.5, Proposition 2.6, and Proposition 4.1 below. Lemma 2.5. Suppose that zero is not a regular point of the spectrum of H = −∆ + V , and let S 1 be the corresponding Riesz projection. Then for sufficiently small λ 1 > 0, the operators M ± (λ) + S 1 are invertible for all 0 < λ < λ 1 as bounded operators on L 2 (R 2 ). Further, one has for any 1 2 and z ∈ C, ℑz = 0), and is a finite-rank operator with real-valued kernel.
Proof. We will give the proof for M + and drop the superscript "+" from formulas. Using Lemma 2.2, Noting that Q ≥ S 1 , we have S 1 P = P S 1 = 0. Therefore, Denote the matrix component of the above equation by A(λ) = {a ij (λ)} 2 i,j=1 . Since Q(T + S 1 )Q is invertible, by the Fehsbach formula invertibility of A(λ) hinges upon the with h(λ) = g(λ) + T r(P T P − P T QD 0 QT P ) = a ln(λ) + z, with a ∈ R and z ∈ C. This follows from (17) and the fact that T r(P T P − P T QD 0 QT P ) is λ independent and real-valued, as the kernels of T , QD 0 Q and v are real-valued. Therefore, d exists if λ is sufficiently small.
Thus, by the Fehsbach formula, Note that S has rank at most two. This and the absolute boundedness of QD 0 Q imply that A −1 (λ) = Finally, we write by a Neumann series expansion.
We now prove the invertibility of the operators 2). Then, in the case of a resonance of the first kind, B ± is invertible on S 1 L 2 (R 2 ) and we have Proof. We again prove the case of the "+" superscripts and subscripts and omit them from the notation. Using Lemma 2.5, we obtain Recall that S 1 D 0 = D 0 S 1 = S 1 . Further, from the definition (21) of S, and the fact that S 1 P = P S 1 = 0, we obtain S 1 SS 1 = S 1 T P T S 1 = T 1 . Therefore Recall that by the definition of a resonance of the first kind, the leading term T 1 in the definition of B is invertible on S 1 L 2 (R 2 ). Therefore, for sufficiently small λ, Combining Lemma 2.4, Lemma 2.5, and Proposition 2.6, we obtain 2). Then in the case of a resonance of the first kind, we have provided that λ is sufficiently small.
Proof. Combining Lemma 2.4, Lemma 2.5, and Proposition 2.6, we have Here we used the fact that Remark. Under the conditions of Corollary 2.7, the resolvent identity holds as an operator identity on , as in the limiting absorption principle, [2].

Resonance of the first kind
In this section, we establish the estimates needed to prove Theorem 1.1. We assume that there is a resonance of the first kind, λ 1 is sufficiently small (so that the analysis in the previous section is valid), and that v(x) x −(1+k)− for k = 1, or equivalently |V (x)| x −4− . It suffices to prove that Theorem 3.1. Under the conditions above, we have for Schwartz functions f and g with f 1 = g 1 = 1.
This theorem will be established in Propositions 3.2, 3.11, 3.12, and 3.13. All statements in this section are valid under the conditions above.
Proposition 3.2. The contribution of the first term in Corollary 2.7 in (1) satisfies (27). More explicitly, we have the bound where p = |x − x 1 |, q = |y − y 1 |, and To prove this proposition, we need to consider the high and low energy contributions of the Bessel functions separately. To this end we use the partitions of unity 1 = χ(λ|y − y 1 |) +χ(λ|y − y 1 |) and We divide the proof of Proposition 3.2 into Lemmas 3.4, 3.8, 3.10 and their respective corollaries, Corollaries 3.5, 3.9, due to the various terms arising in (28).
For the low energy parts, the following lemma will be useful: Then for any τ ∈ [0, 1] and λ ≤ 2λ 1 we have Here k(x, x 1 ) : Proof. We start with G. Let g(s) := χ(s)J 0 (s). We have g ′ (s) = O(1). Therefore, by the mean value theorem and the boundedness of g, we have for any 0 ≤ τ ≤ 1.

Now consider
Let g 1 (s) = sg ′ (s). We have |g 1 (s)| 1 and |g ′ 1 (s)| 1. Therefore, by the mean value theorem and the boundedness of g 1 , we have The bounds for F were obtained in [23]. We repeat them for completeness. Note that F (0+, x, By inspecting the integrands on the right hand side, we see that |∂ λ F | is bounded by 1/λ. To obtain the statement for |F | first note that, since χ ′ is supported in the set [λ 1 /2, 2λ 1 ], the first line in (29) is 1. To estimate the second line note that Lemma 3.4. We have the bound Proof. Since S 1 ≤ Q are projections and Q is the projection orthogonal to v, we have for all h ∈ L 2 (R 2 ). As such, we can subtract functions of x (resp. y) only from χY 0 (resp. χJ 0 ) in the integrand of (31). We use the functions defined in Lemma 3.3. Thus we replace We integrate by parts once to get There is no boundary term since, by Lemma 3.3, we have that F (0+, y 1 , y) k(x, x 1 ) and G(0, y, y 1 ) = 0. From Lemma 3.3 again, we have for any τ ∈ (0, 1] Taking τ = 0+, this term now contributes the following to (31), For the case of (35) and (36), we again note the bounds in Lemma 3.3, and that on the support of χ(λ), |λ τ log λ| 1 for any τ > 0. The desired bound follows as in (37).
We also need the following bounds taking care of the contributions of the remaining terms in (28): Proof. Using the notation of Lemma 3.4, we need to bound and the similar term when F is replaced by G. This follows easily from one integration by parts and the bounds of Lemma 3.3 as in the previous lemma.
We now need to bound the resulting terms when one of the Bessel functions is supported on large energies. The following variation of stationary phase from [23] will be useful in the analysis. For completeness we give the proof.
In addition we have the following high-energy analogue of Lemma 3.3. In light of the high energy representations of the Bessel functions (9), recall that for C ∈ {J 0 , Y 0 , H 0 }, with ω ± as in (9). Then for any 0 ≤ τ ≤ 1 and λ ≤ 2λ 1 , Proof. We note first that from (9), we have We consider the case of G + , the case of G − is similar. Define the function b(s) := χ(s)ω + (s).
Using (9), one obtains that for k = 0, 1, 2, ..., We now rewrite G in terms of b: Note that the absolute value of the last summand is To estimate the difference of the first two we assume without loss of generality that p > q and write In the case, 1 < λq < λp, we estimate this integral by In the case λq < 1 < λp, we estimate it as follows λp λq χ(s)|s| − 3 2 ds χ(λp) Combining these bounds and interpolating with (40) we obtain the first assertion of the lemma.
We now turn to the derivative. We note that where b 1 (s) := sb ′ (s) satisfies the same bounds that b(s) does. Therefore the second assertion of the lemma follows as above.
Lemma 3.8. We have the bound Proof. Without loss of generality we assume that t > 0. As in the proof of the previous statements, it suffices to prove that for fixed x, x 1 , y, y 1 the λ-integral is bounded by k(x, x 1 ) y 1 t −1 . This power of y 1 necessitates extra decay on the potential to push through the L 2 mapping bounds as in the previous lemmas. Accordingly, we assume that v(x) Let p = max(|y − y 1 |, 1 + |y|) and q = min(|y − y 1 |, 1 + |y|). Using (32), it suffices to consider where F (λ, x, x 1 ) is as in Lemma 3.3. The oscillatory term in the definition (9) of J 0 for large energies will move the stationary point of the phase. Pulling out the slower oscillation e ±iλq , we rewrite this integral as a sum of where φ ± (λ) = λ 2 ± λqt −1 , and G is from Lemma 3.7. Note that this moves the stationary point of We first consider the contribution of the term with the phase φ − (λ) in which case the critical point Using the bounds in Lemma 3.3 and Lemma 3.7 (with τ = 0+), we have 1 2 , and (42) We now apply Lemma 3.6 with a(λ) as above to bound the λ-integral in this case by Using (42), we bound the first integral in (44) by There are two cases: In the former case, on the support of the integral, we have λ λ 0 . Therefore, In the last inequality, we used p −1 λ 0 ≤ q −1 λ 0 ≤ t −1 . In the latter case, on the support of the integral, For the χ(λp) term to have any contribution to the integral, we must have that p −1 t − 1 2 , similarly for q −1 . So that, It suffices to bound the second integral in (44) by k(x, x 1 ) y 1 . We first establish the bounds for the a(λ) term and then consider the derivative a ′ (λ).
For the portion of a(λ) supported on On the other hand if λ 0 t − 1 2 , we have Fix i, j and let m = min(p i , q j ). We have two cases: λ 0 ≪ 1/m and λ 0 1/m. In the former case, we note that |λ − λ 0 | 1/m on the support of the cutoffs. Therefore, In the latter case, using (52), we conclude that p i q j t. This implies the desired bound by ignoring the cutoffs in the integral.
We now turn to the term in Lemma 3.6 that involves a ′ (λ). Using (50), we have The required bounds for each of these terms appeared above in the bound for a(λ)/|λ − λ 0 | 2 integral.
This establishes the desired bound for the phase φ − . For the case of φ + , integration by parts and the bounds on a(λ) and a ′ (λ) suffice, we leave the details to the reader.
With these estimates established, we are ready to prove Proposition 3.2.
We now turn to the terms involving SS 1 D 1 S 1 and S 1 D 1 S 1 S in Corollary 2.7.
Proposition 3.11. The contribution of the terms SS 1 D 1 S 1 , S 1 D 1 S 1 S and QD 0 Q in Corollary 2.7 in (1) satisfies (27). More explicitly, we have the bound The same bound holds when SS 1 D 1 S 1 is replaced by S 1 D 1 S 1 S or by QD 0 Q.
Proof. The QD 0 Q term can be handled as in Proposition 3.2, it is in fact easier since there is no log(λ) term.
The other terms are somehow different since they have a projection orthogonal to v only on one side. Therefore, one can use (32) only on one side. However, since there is no log(λ) term, the bounds established in in Lemmas 3.4, 3.8, and 3.10 go through. For instance, to establish the bound we can use G(λ, y, y 1 ) in place of J 0 (λ|y − y 1 |). After an integration by parts the boundary terms vanish since G(λ, y, y 1 ) → 0 as λ → 0, and the λ-integral can be bounded by Here we used the bounds for G from Lemma 3.3, the bounds (6) and (7), and the following estimate: This estimate follows easily by considering the cases |x − x 1 | < 1 and |x − x 1 | > 1 separately.
When we have S 1 D 1 S 1 S instead, we must use F (λ, x, x 1 ) instead of Y 0 (λ|x−x 1 |), and the boundary Similarly, in the case when χ is replaced with χ on the side which does not have a projection orthogonal to v, the proof of Lemma 14 from [23] applies.
It remains to prove that Since S is not orthogonal to v, we can not replace χY 0 with F . However, we can replace χJ 0 with G shifting the critical point of the λ-integral as in the proof of Lemma 3.8. The argument in the proof of that lemma relies on the bounds (55) | log(λ)F (λ, x, x 1 )| log(λ)k(x, x 1 ), ∂ λ log(λ)F (λ, x, x 1 ) k(x, x 1 ) log(λ)λ −1 .
The terms arising from h ± (λ) −1 S and h ± (λ) −1 SS 1 D 1 S 1 S are handled in Lemma 17 in [23], which we restate below for completeness.
A similar bound holds if we replace S with SS 1 D 1 S 1 S.
Finally the following proposition (Lemma 18 from [23]) takes care of the contribution of the error term in Corollary 2.7 to (1).

Resonances of the second and third kind
We now consider the evolution in the case of a p-wave resonance and/or an eigenvalue at zero.
Recall that this case is characterized by the non-invertibility of T 1 = S 1 T P T S 1 . To obtain resolvent expansions around zero, we need to invert the operator B ± , (19). The expansions in this section are considerably more complicated than those in the case of a resonance of the first kind given in Proposition 2.6.
Recall the operators S 2 , S 3 , T 2 , and T 3 from Definition 2.3. With a slight abuse of the notation, we define D 1 := (T 1 + S 2 ) −1 = (S 1 T P T S 1 + S 2 ) −1 as an operator on S 1 L 2 (R 2 ). We define D 2 := i.e. when S 2 = 0. We also define D 3 := T −1 In the case of a resonance of the second kind, we have where g ± 1 (λ) is as in Lemma 2.2. In the case of a resonance of the third kind, we have Proof. We give the proof for the case of the "+" superscripts and subscripts and omit them from the proof. Recall the definition (19) of B: First we repeat the expansion that we obtained in Proposition 2.6 by keeping track of the error term better. Using Lemma 2.5, the identity and the definition (22) of A −1 (λ), we obtain where (22)), we conclude that Recall that S 1 D 0 = D 0 S 1 = S 1 . Further, from the definition (21) of S, and the fact that S 1 P = P S 1 = 0, we obtain S 1 SS 1 = S 1 T P T S 1 . Therefore In the case of a resonance of the second kind (unlike the case of a resonance of the first kind), the leading term T 1 = S 1 T P T S 1 above is not invertible. We will invert the operator by using Lemma 2.4. Let S 2 be the Riesz projection onto the kernel of T 1 , and let D 1 := (T 1 + S 2 ) −1 .
We have By Lemma 2.4, B 1 is invertible if is invertible on S 2 L 2 . Using (62), the identities S 2 D 1 = D 1 S 2 = S 2 , and the definition (59) of E(λ), we have We now claim that To see this, note that since S 2 , T and P are self-adjoint, and S 2 is the projection onto the kernel of Therefore, Using this and the expansion (14) of M 0 , we rewrite B 2 as (1) and (60), we conclude that E 1 (λ) = O 1 (λ 4− ). This yields that E 2 (λ) = O 1 (λ 2− ). In the case of a resonance of the second kind the leading term is invertible. Therefore, for small λ, Using Lemma 2.4, (62), (66), and the identities S 2 D 1 = D 1 S 2 = S 2 , we have In the case of a resonance of the third kind, the leading term in B 2 is not invertible. Analogously, we will invert the operator by using Lemma 2.4. Let S 3 be the Riesz projection onto the kernel of T 2 , and let D 2 := (T 2 + S 3 ) −1 .
We have In the second line we used the definition of g 1 (λ) in Lemma 2.2 and the estimate on E 2 (λ).

By Lemma 2.4, B 3 is invertible if
Since T 3 is always invertible (see Section 4 of [14]), B 4 is invertible for small λ, and we have Using this, Lemma 2.4, and (68), we have Using this (instead of (66)) for B −1 in (67) yields the assertion of the proposition.
where Γ ± i , i = 1, 2, 3, 4 are absolutely bounded operators on , and Γ ± 4 (λ) = O(λ −2 (log λ) −4 ). In the case of a resonance of the third kind, we have where D is as in Proposition 4.1, and Γ i are absolutely bounded operators on L 2 (R 2 ). These operators are distinct from the Γ i in the case of a resonance of the second kind, but satisfy the same size estimates.
Proof. For a resonance of the second kind, combining Proposition 4.1 with Lemma 2.4 and Lemma 2.5 (taking the decay condition on v into account), we have

Using (65) and the definition (22) of
The second line leads to four different terms yielding (69).
For the case of a resonance of the third kind, the statement follows similarly using the formula (57) for B −1 .
We now consider the dispersive estimates in the case when H has a p-wave resonance at zero energy. Comparing (69) to the expansion in Corollary 2.7, we note that the many of the terms in the expansion for resonances of the second kind are in the expansion for resonances of the first kind.
Accordingly, it suffices to establish the estimates for the contributions of the terms: We start with the following.
Proof. We note that we must exploit some cancellation between the '+' and '−' terms. Recall that H ± 0 (y) = J 0 (y) ± iY 0 (y) and the definition of g ± 1 (λ) in Lemma 2.2 give us We again must use the cut-offs χ and χ and consider the different cases depending the supports of the resolvents. Let us first consider the case when both resolvents are supported on low energy.
Contribution of the first term in (72) satisfies the required bound since J 0 = O(1), and 1 λ(log λ) 2 is integrable on [0, λ 1 ]. Since the other terms have additional powers log λ in the numerator, we need to use (32) (recall that S 2 ≤ Q).
Consider the contribution of the second term in (72). Using (32), we replace χY 0 with F (λ, ·, ·), and using Lemma 3.3, we obtain the bound: The mixed J 0 and Y 0 terms in the second part of (72) are bounded similarly using |G(λ, x, An analysis as in (37) shows that these terms satisfy the desired bound (71).
When one or both of the Bessel functions is supported on high energies, we use the functions G(λ, p, q) from Lemma 3.7. The bound | G(λ, p, q)| λ 0+ |p − q| 0+ suffices for obtaining the required bound. The details are left to the reader.
Lemma 4.4. For C i (z) = J 0 (z) or Y 0 (z) for i = 1, 2, we have the bound Proof. Unlike in Lemma 4.3 we do not need to use any cancellation between the '+' and '−' terms.
We consider the terms that arise when both C 1 and C 2 are supported on small energies. Consider, where p = |x − x 1 |, q = |y − y 1 |. In the worst case when C 1 = C 2 = Y 0 , using (32), we replace χY 0 with F to obtain The last line follows from Lemma 3.3. Since sup 0<λ<λ1 |λ 2 (log λ) 2 Γ 1 (λ)| defines a bounded operator on L 2 (R 2 ) (by Corollary 4.2), we are done. The other low energy terms are similar using G instead of F from Lemma 3.3.
For the large energies, we note that the argument runs in a similar manner. Using χ(y)(|J 0 (y)| + |Y 0 (y)|) 1, and an argument as in (73), it easily follows that the integral is bounded as desired.
The following modification of Lemma 4.4 is necessary for the other Γ i (λ) terms.
The same bounds hold when QΓ 2 (λ) is replaced by Γ 3 (λ)Q or Γ 4 (λ. Proof. We repeat the analysis of Lemma 4.4. Consider the case when both C i (λ·) are supported on low energies and both are Y 0 . We note that when λ < 1, using (53), we have Using this and replacing χY 0 with F on one side, we obtain the bound The same bound holds for Γ 3 (λ)Q. For the contribution of Γ 4 (λ), we have The other cases are similar.
When one of the C i (λ·) is supported on high energies, the analysis is less delicate. The required bound follows from χ(y)(|J 0 (y)| + |Y 0 (y)|) 1.
This completes the proof in the case of a resonance of the second kind.
We note that the above bounds in Lemma 4.4 and Corollary 4.5 also hold for the Γ i term in (70).
Thus for a resonance of the third kind, it suffices to consider the leading λ −2 term in (70). Noting (28) and the fact that the kernel of D 3 is real-valued, the following lemma completes the proof. We will prove in the next section that G 0 vS 3 D 3 S 3 vG 0 is the projection onto the zero eigenspace whose contribution disappears since we project away from the zero eigenspace. We will ignore this issue in the proof below since the eigenfunctions are bounded functions and hence the projection onto the zero eigenspace satisfies the desired bound, and since removing this operator requires more decay from the potential, see Section 6.
Lemma 4.6. We have the bound Proof. We provide a sketch of the proof. Due to similarities to previous proofs, we leave the details to the reader. We again consider the case when the Bessel functions are supported on low energy first.
Accordingly, we wish to control Where we used (32), Lemma 3.3 with any τ > 0.
For the case when one function is supported on high energy, we have Similarly one uses G(λ, y, y 1 ) instead of F (λ, y, y 1 ) if we have χ(λq).
When both functions are supported on high energy, we have An analysis as in (37) finishes the proof.
We are now ready to prove the main theorem of this section.
Theorem 4.7. Let V : R 2 → R be such that |V (x)| x −β for some β > 6. Further assume that H = −∆ + V has a resonance of the second or third kind at zero energy. Then, there is a time dependent operator F t such that Proof. If we denote the terms that arise from the contribution of the terms in the first lines of (69) and (

Spectral Structure of −∆ + V at Zero Energy
In this section, we prove some of the claims made in the remark following Definition 2.3. In particular, we show the relationship between the spectral subspaces S i L 2 (R 2 ) for 1 = 1, 2, 3 and distributional solutions to Hψ = 0.
Let w = U v. First we characterize S 1 L 2 .
Now we prove that ψ ∈ L ∞ . The boundedness on B(0, 4) is clear. To see that ψ is bounded for |x| > 4, use P φ = 0 to obtain The bound follows by using the inequality (for |x| > 4) Note that this only requires that v(x) x −1− .
The final statement follows if we can prove that G 0 vφ = O(|x| −1 ) for large x. To see this, write The first integral can be estimated by On the other hand, the bound for the second integral follows from Let v x −2− . Assume that the function ψ = c + Λ 1 + Λ 2 , with Λ 1 ∈ L p , for some p ∈ (2, ∞), and Λ 2 ∈ L 2 , solves Hψ = 0 in the sense of distributions. Then φ = wψ ∈ S 1 L 2 , and In particular, by the previous claim, ψ − c ∈ L p for any p ∈ (2, ∞].
Since ψ + G 0 vφ ∈ L 2 + L ∞ (by assumption and the proof of the previous claim), we see that it has to be a constant. Thus Using this, we have and hence QT Qφ = 0, and φ ∈ S 1 L 2 . Finally, this implies that c = 1 V L 1 v, T φ .
Note that Lemma 5.1 and Lemma 5.2 imply that all zero eigenfunctions are bounded. We now characterize S 2 L 2 . Proof. Recall that S 2 ≤ S 1 is projection onto the kernel of S 1 T P T S 1 . We have (since S 1 φ = φ) and hence c 0 = 0 and ψ ∈ L p for p > 2.
On the other hand if ψ ∈ L p for p > 2, we have c 0 = 0. This implies that P T φ = P T S 1 φ = 0 ,and hence S 1 T P T S 1 φ = 0.

Lemma 5.4.
If v x −3− then the kernel of the operator S 3 vG 2 vS 3 on S 3 L 2 is trivial.
Proof. Given f in the kernel of S 3 vG 2 vS 3 , we have Also note that the expansion we used for R + 0 (λ 2 ) in the proof of Lemma 2.2 gives that This and the assumption v x −3− imply that Now, using (78), we have Where we used the monotone convergence theorem in the last step. By the assumptions on v and f , vf ∈ L 1 , and hence vf = 0. We also know that f ∈ S 1 L 2 and hence f = wψ, which implies that f = 0. This establishes the invertibility of the operator S 3 vG 2 vS 3 on S 3 L 2 .
It remains to prove the claim above. In what follows below we can assume that |x| > 4 since ψ ∈ L ∞ . Define the set B := {y ∈ R 2 : |y| < |x|/8}. Recall that we have ψ = −G 0 vφ, and as P φ = 0 we have First we note that the second term is in L 2 . Indeed, using (76), and then 1 y / x , we see that We now examine the integral on B. We note that on B, |y| 2 − 2x · y /|x| 2 < 1 2 , and hence ln |x − y| 2 The error term is in L 2 . We also note that Therefore, we can rewrite the main term as Using this in (81), we obtain yv(y)φ(y) dy with Ψ ∈ L 2 . As x/|x| 2 is not in L 2 (R 2 ), we must have (80). Proof. Let {φ j } N j=1 be an orthonormal basis for the S 3 L 2 , the range of S 3 . Then, we have We have φ j = wψ j for each j with ψ j ∈ L 2 . Since P S 3 = 0, we also have Since {φ j } N j=1 is linearly independent, we have that {ψ j } N j=1 is linearly independent, and it follows from (82) that Using the orthonormal basis for S 3 L 2 , we have that for any f ∈ L 2 (R 2 ), Let A = {A ij } N i,j=1 be the matrix representation of S 3 vG 2 vS 3 with respect to the orthonormal basis of S 3 L 2 . Using (79), Let P e := G 0 vS 3 [S 3 vG 2 vS 3 ] −1 S 3 vG 0 . Then by (83), for any f ∈ L 2 (R 2 ), Note that for f = ψ k , 1 ≤ k ≤ N , Thus, we can conclude that the range of P e is equal to the span of {ψ j } N j=1 and that P e is the identity on the range of P e . Since P e is self-adjoint, the claim is proven.

A Weighted Estimate
In this section we prove Theorem 1.2. Recall that if zero is an eigenvalue but there are neither s-wave nor p-wave resonances at zero, then S 1 = S 2 = S 3 = 0. We note that in this case many terms in the expansions of M ± (λ) −1 in Corollaries 2.7 and 4.2 disappear. This follows as now (84) P S 1 = S 1 P = 0, S 1 T P = P T S 1 = 0, S 1 vG 1 vS 1 = 0.
We will also need a finer expansion for M 0 (λ) then it is given in Lemma 2.2 to prove the theorem.
Define g ± 2 (λ) = λ 4 (a 2 log λ + b 2,± ) and g 3 (λ) = a 3 λ 4 with a 2 , a 3 ∈ R \ {0} and b 2,− = b 2,+ . Also let G 3 be the integral operator with the kernel |x − y| 4 , and G 4 with the kernel |x − y| 4 log |x − y|. Similar to the expansion given in Lemma 2.2 we obtain by expanding the Bessel functions to order z 6 log z and estimating the error term as in Lemma 2.2.
Using this and (84) in (22) and (85), we have Therefore, using these expansions in (88), we have where Γ i are absolutely bounded operators on L 2 with Γ i = S 3 Γ i S 3 , and Γ −1 1 = D 3 . Inverting this via Neumann Series yields the claim of the proposition. Corollary 6.2. Assume that S 1 = S 2 = S 3 , and that |V (x)| x −11− . Then Here, Ξ i are real-valued absolutely bounded operators, Ξ 2 and Ξ 3 have a projection orthogonal to P on at least one side, and Ξ 1 have orthogonal projections on both sides. Further a i ∈ R \ {0} and We should note that in the statement of the corollary we listed only one term of each form. For example there are several different terms of the form b3,± a2 log λ+b2,± Ξ 2 in the expansion.
Using Corollary 6.2 and Lemma 5.6 in (26), we see that the contribution of the D 3 /λ 2 term can be written as In the first line above, we used the fact that P D 3 = D 3 P = 0 to subtract off g + (λ).  Proof. First note that since we project away the zero eigenspace, the contribution of (93) cancels out.
For the contribution of the terms in (92), we need to use the cancellation between the '+' and '-' terms in Stone's formula.
Here we used that on the support of χ(λp), we have 1 λp. At this point, the proof follows exactly along the lines of Lemma 3.8 with the extra weights of p + q y y 1 , which yields the required bound.
We are now ready to prove the theorem. We provide a sketch, as there is a significant overlap with the proofs of previous estimates in Section 3.
Proof of Theorem 1.2. We already proved the theorem for the contribution of the D 3 /λ 2 term in Corollary 6.2 to (26). The contribution of the Ξ 1 term and the terms in the second line of (90) satisfies the dispersive bound by the results of Section 3. It remains to control the contribution of 1 + b 3,± a 2 log λ + b 2,± Ξ 2 .
We will only provide a brief sketch. Recall that Ξ 2 has projection orthogonal to P only on one side, say on the right. On high energy, we can use λ|x − x 1 | 1 to extract positive powers of λ for the integration at the loss of a weight as in the proof of Lemma 6.3. The polynomial weights arising are either ameliorated by the decay of the potential v or goes into the weight of the weighted dispersive bound. For the low energy part, the worst case is when we have Y 0 on both sides. This arises only with the term containing log λ in the denominator due to the cancellation between the ± terms. On the right hand side, using (32), we replace χY 0 with F from Lemma 3.3 to reduce to bounding the following integral ∞ 0 e itλ 2 λχ(λ)χ(λ|x − x 1 |)Y 0 (λ|x − x 1 |) b 3 a 2 log λ + b 2 F (λ, y, y 1 ) dλ .
We note, from (5), that The first log λ is the most troubling, we note that to control it we use the following facts The contribution of the other terms can be bounded by similar arguments.