Resolvent bounds for Lipschitz potentials in dimension two and higher with singularities at the origin

We consider, for $h,E>0$, the semiclassical Schr\"odinger operator $-h^2\Delta + V - E$ in dimension two and higher. The potential $V$, and its radial derivative $\dell_{r}V$ are bounded away from the origin, have long-range decay and $V$ is bounded by $r^{-\delta}$ near the origin while $\dell_{r}V$ is bounded by $r^{-1-\delta}$, where $0\leq\delta\leq 4(\sqrt{2}-1)$. In this setting, we show that the resolvent bound is exponential in $h^{-1}$, while the exterior resolvent bound is linear in $h^{-1}$.

The operator P (h), with the potential V satisfying (1.2) to (1.7), is self-adjoint with respect to the domain D(P ) = H 2 (R n ) [Nel64].
The interest in resolvent estimates of the form (1.9) can be traced back to Burq [Bur98], who showed (1.9) in the case of smooth, compactly supported potentials.The resolvent estimates were then used to estimate the rate of decay of the local energy of the wave equation when an obstacle was present in the domain.Following works by Vodev, Burq and Cardoso and Vodev [Vod00,Bur02,CV02] expanded on the resolvent estimates found in [Bur98], with [Vod00] generalising the estimate to a class of noncompactly supported potentials, [Bur02] extends the estimates in [Bur98] to the case of smooth, long-range potentials, and Cardoso and Vodev [CV02] refine Burq's work [Bur02] to give exterior estimates of the form seen in (1.10).
Datchev's work [Dat14] provided resolvent estimates (1.9) and (1.10) for potentials with weaker assumptions placed on their regularity in dimensions n = 2.The only requirement was that the potential V and its radial derivative ∂ r V be bounded and satisfy certain decay estimates.Datchev used an energy functional to prove global Carleman estimates, a technique which features in later works on resolvent estimates, including this paper.Further works that give resolvent estimates with little regularity assumed are [Vod14, RT15, DdH16, KV18, Vod19, Sha20, Vod20a, Vod20b, Vod21a, Vod21b, Vod22].
Of particular importance to this paper is Shapiro's work [Sha19], which shows (1.9) and (1.10) for long-range potentials V in two dimensions, expanding on [Dat14] to include the two dimensional case.However, [Sha19] requires decay conditions on the whole gradient ∇V while [Dat14] only requires a derivative in the radial variable.This paper extends the results of [Dat14] to the two dimensional case, in particular, requiring only assumptions on ∂ r V , rather than ∇V .This paper is a continuation of the joint work of Galkowski and Shapiro [GS22], who gave a proof for resolvent estimates in the case of potentials that are unbounded at the origin with long-range decay.We prove resolvent estimates for potentials with singularities at the origin in two dimensions and higher, extending the results of [GS22] to a class of potentials with greater growth at the origin.Notably, the resolvent estimates are true in the case of Coulomb potentials in three dimensions.We will see that, in order to handle singularities in a soon-to-be-defined energy functional at the origin, we require similar methods to those used by Galkowski and Shapiro in [GS22] with the added use of the Mellin transform inspired by [DGS23].In [DGS23], these Mellin transform methods were used to prove resolvent estimates in the case of compactly supported radial potentials that are only L ∞ .Here, we use these methods to handle large singularities as well as the case of dimension two simultaneously.
The approach taken to prove Theorem 1 involves defining the conjugated operator where the phase function ϕ is absolutely continuous and defined on [0, ∞) with ϕ ≥ 0, ϕ(0) = 0 and ϕ ′ ≥ 0 and ∆ S n−1 is the Laplace-Beltrami operator on S n−1 .Throughout this paper, integrations are carried out with respect to the measure drdθ and the Lebesgue measure on R n .To avoid confusion, we will distinguish between [0, ∞) × S n−1 and R n , with integration that takes place on subsets of [0, ∞) × S n−1 being done with respect to drdθ and integration that takes place on R n being done with respect to the Lebesgue measure.
We now define an energy functional where we drop the L 2 (S n−1 θ ) subscript for ease of notation.
In the proof of Theorem 1 we attain a lower bound for (wF ) ′ by constructing appropriate weight and phase functions w and ϕ.This requires extra work in dimension two compared to dimensions n = 2 because of the (2wr −1 − w ′ ) h 2 r −2 Λu, u term in (1.15).In dimensions n = 2, the operator Λ is non-negative, so this term can be bounded below by zero, provided we require 2wr −1 − w ′ ≥ 0. In dimension two, however, Λ has a negative eigenvalue and the rest of the eigenvalues are positive.This means a negative singularity occurs at r = 0, which requires extra care when compared to the cases when n = 2.

Near Origin Estimates
The first step is to establish control on the behaviour of u near the origin.Much of this uses techniques used in Section 4 [DGS23], but adapted for the more general n dimensional problem with potentials that need not be either L ∞ , compactly supported or radial.
We define an integral transform, M, known as the Mellin transform, and its inverse M −1 t by Re σ,θ (R × S n−1 ) respectively, the Mellin transform and its inverse are well-defined, bounded operators M : The Mellin transform and its inverse satisfy the following for all θ ∈ S n−1 : where σ = τ + it ∈ C and Γ(z) is the Gamma function [DB16].
We now define λ j := j 2 + (n − 2)j + (n−1)(n−3) 4 for j ∈ N 0 , which are the eigenvalues of Λ.Let which is the set of the imaginary parts of the poles of the map σ → (σ 2 − iσ + Λ) −1 .Elements of T are of the form we have the following result.
Lemma 2.1.There exists a C > 0 such that, for where for each θ ∈ S n−1 and Proof.Given t 0 ∈ R \ T , we can assume, without loss of generality, that We want to deform the contour to Im σ = t 0 − ε for ε > 0 then take the limit as ε → 0.
In the region Im We can therefore define (2.8) Using (2.1), we can write We now show that for Im σ = t 0 , (σ 2 − iσ + Λ) −1 on L 2 (S n−1 ) is bounded by Υ(t 0 ).As Λ is self-adjoint, we have that To find an upper bound on dist(iσ − σ 2 , {λ j : j ∈ N 0 }) −1 , we notice that sup Through calculation it can be showed that the minimum of |iσ − σ 2 − λ j | with respect to Re σ occurs when Re σ = 0, therefore sup Now, using (2.2) and (2.3) we obtain the desired result.
To calculate Π t 0 (r 2 Qχ α 1 u), we refer to (2.8).We need to find where the singularities of r −iσ (σ 2 − iσ + Λ) −1 M(r 2 Qχ α 1 u) occur in the σ variable, which is done by solving for σ j in σ 2 j − iσ j + λ j = 0 for −N − 1 < Im σ j < t 0 for each j ∈ N 0 .In doing so we see that σ j = it j , where t j ∈ T can be written in the form given by (2.5).We also require −N − 1 < Im σ j < t 0 , so each of the σ j have a negative imaginary part.Therefore, there are a finite amount of σ j being summed over, with the form σ j = i 1 2 (3 − n − 2j), where j ≥ 0 for n ≥ 3 and j ≥ 1 when n = 2.The Θ j ∈ L 2 (S n−1 ) are given by therefore we can write, for some J ∈ N, where j 0 = 1 when n = 2 or n = 3 and j 0 = 0 otherwise.For κΠ t 0 (r 2 Qχ α 1 u) to be in L 2 (R + × S n−1 ), we must have Π t 0 (r 2 Qχ α 1 u) = 0, in particular Using this fact, Lemma 2.1 and (2.10), we arrive at . (2.12) We have defined χ α 1 so that it is supported on the interval [0, 2α 1 ] and bounded above by 1, therefore .
Using (3.7) and the fact that W = 1 2 r implies q = 0 and that h 0 < 27E 4(1+η)K , we get , which, together with We defined M to be greater than or equal to max{12 √ 3, 16KE − 1 2 }, which gives us On the region r ≥ M , we have ϕ ′ = 0, which reduces (3.7) to By making the substitution W ≤ Er 3 4(c 0 r 2 m(r)+1) we see that

Carleman Estimates
To be able to prove Theorem 1, we first give a Carleman estimate.We begin by proving the following lemma.
Lemma 4.1.There are constants C, h 0 > 0 that are independent of h and ε so that for all ε > 0 and h ∈ (0, h 0 ], and for all u ∈ r The proof of this lemma follows a similar argument to that can be found in the proof of Lemma 3.2 in [GS22], but is adapted for the use of a weight function w that is quadratic near the origin.
Proof.Starting with (3.2) and applying Lemma 3.1, for h ∈ (0, h 0 ] we get where the norm and inner product used in this inequality are those of the space L 2 (S n−1 θ ), and η > 0 depends on δ, as implied by Lemma 3.1.
Integrating the inequality above with respect to r from 0 to ∞ and using the fact that wF, (wF ) ′ ∈ L 1 (0, ∞) and w(0) = 0, we get ´∞ 0 (wF ) ′ dr = 0. From (3.5) and (1.8) we see that W −1 ∈ L 1 ((0, ∞)), which implies the boundedness of w.This, together with the fact that We now use the Cauchy-Schwarz inequality on 2ε ´r,θ w|uu ′ |, which gives where c > 0 is the implicit constant in (4.2).Then for 0 < r < β, we have cε h w ≤ 1 2 w ′ .The term ´r≤β w|hu ′ | 2 can be subtracted from both sides of (4.2), giving ˆr,θ where we have used the fact that w is bounded.Using the facts that on supp ψ, |V − E − (ϕ ′ ) 2 | and r −2 are bounded and r −2s w ′ and that Λ ≥ − 1 4 together with the two equations above, we have that, for all h ∈ (0, 1] and γ > 0, ˆr,θ After substituting (4.4) into (4.3),we choose γ > 0 small enough, and then making h 0 sufficiently small, giving ˆr,θ We are now in a position to prove the Carleman estimate.
Lemma 4.2.There are constants C 1 , C 2 > 0 independent of h and ε such that for all ε > 0 and h ∈ (0, h 0 ], and for all v ∈ C ∞ c (R n ).Here, the measure used to define the L 2 norms in use in (4.5) and (4.6) is the Lebesgue measure on R n .
Proof.We begin with the proof by showing the inequality (4.5) holds.We start by defining u = r The second term, which describes u away from the origin, can be estimated as follows for some constant C > 0 that is independent of h and ε.We have dropped the drdθ for ease of notation.As for the first term, we have the estimate where t 0 ∈ (− 1 2 , 0).Combining (4.8) and (4.9) with (4.7) gives ˆr,θ We now look towards Lemma 2.2 to turn (4.10) into an estimate in terms of (P − E ± iε)u and u, r Next, we make use of (1.3) and that w ′ ∼ r on A(a, 2α 1 ) to arrive at Furthermore,since t 0 < 0, we have From Lemma 2.2 we have . (4.14) 14 For small enough h 0 > 0, a ∼ h 2 2−δ for h ∈ (0, h 0 ], therefore h −4 ∼ a −2(2−δ) , so a 1+2t 0 h −4 ∼ a −3+2t 0 −2δ .Substituting inequalities (4.11) to (4.13) into (4.14)allows us to obtain Because t 0 > − 1 2 and δ ≥ 0, we get a −3+2t 0 +2δ ≤ a −4 and therefore From here, we make use of (4.10) followed by the substitution a ∼ h 2 2−δ and arrive at We now use Lemma 4.1 and a substitution u = r n−1 2 v to arrive at (4.5).To show (4.6), we use the observation that r −2s w ′ for r ≥ M , which implies then the use of Lemma 4.1 yields where C ϕ := 2 max ϕ.Furthermore, 2ϕ(r) = C ϕ for r ≥ M because ϕ ′ ≥ 0 for all r > 0 and ϕ ′ = 0 for r ≥ M .Dividing through by e Cϕ h gives us (4.6).

Resolvent Estimates
The goal of this section is to prove Theorem 1.This proof follows the same argument as used in Section 5 of [GS22].
The operator [P, r s ] r −s = (−h 2 ∆ r s − 2h 2 (∇ r s ) • ∇) r −s is bounded H 2 → L 2 , so, for all u ∈ H 2 (R n ) such that r s u ∈ H 2 (R n ), for some constant C ε,h depending on ε and h.Given f ∈ L 2 (R n ), the function u = r s (P − E ± iε) −1 r −s f ∈ H 2 (R n ) because u = (P − E ± iε) −1 (f − w), where w = r s [P, r −s ] r s r −s u is L 2 because the operator r s [P, r −s ] r s is bounded H 2 → L 2 since s < 1 and r −s u = (5.6) Also, applying (5.5) gives (5.7) We then replace u by ũk in (5.3) and send k → ∞ to attain (5.4).