Abstract
We prove semi-classical resolvent estimates for real-valued potentials V ∈ L ∞ (R n ), n ≥ 3, of the form V = VL + VS, where VL is a long-range potential which is C 1 with respect to the radial variable, while VS is a short-range potential satisfying VS(x) = O x −δ with δ > 1.
Introduction and statement of results
The goal of this paper is to extend the semi-classical resolvent estimates obtained recently in [7] , [9] and [11] to a larger class of potentials. We are going to study the resolvent of the Schrödinger operator
Introduction and statement of results
where 0 < h ≪ 1 is a semi-classical parameter, ∆ is the negative Laplacian in R n , n ≥ 3, and V ∈ L ∞ (R n ) is a real-valued potential of the form V = V L + V S , where V L ∈ C 1 ([r 0 , +∞)) with respect to the radial variable r = |x|, r 0 > 0 being some constant, is a long-range potential, while V S is a short-range potential satisfying (1.1) |V S (x)| ≤ C 1 (|x| + 1) −δ with some constants C 1 > 0 and δ > 1. We suppose that there exists a decreasing function p(r) > 0, p(r) → 0 as r → ∞, such that (1.2) V L (x) ≤ p(|x|) for |x| ≥ r 0 .
Introduction and statement of results
We also suppose that (1.3) ∂ r V L (x) ≤ C 2 (|x| + 1) −β for |x| ≥ r 0 with some constants C 2 > 0 and β > 1. As in [11] we introduce the quantity g ± s (h, θ) := log (|x| + 1) −s (P (h) − E ± iθ) −1 (|x| + 1) −s
Introduction and statement of results
where L 2 := L 2 (R n ), 0 < θ < 1, s > 1/2 is independent of h and E > 0 is a fixed energy level independent of h. Our first result is the following Then there exist constants C > 0 and h 0 > 0 independent of h and θ but depending on s and E such that for all 0 < h ≤ h 0 we have the bound (1.5) g ± s (h, θ) ≤ Ch −4/3 log(h −1 ).
Introduction and statement of results
When V S ≡ 0 and V L satisfying conditions similar to (1.2) and (1.3) , it is proved in [4] when n ≥ 3 and in [8] when n = 2 that (1. 6) g ± s (h, θ) ≤ Ch −1 with some constant C > 0 independent of h and θ. Previously, the bound (1.6) was proved for smooth potentials in [2] and an analog of (1.6) for Hölder potentials was proved in [10] . A high-frequency analog of (1.6) on Riemannian manifolds was also proved in [1] and [3] . When V L ≡ 0 and V S satisfying the condition (1.1) with δ > 3, the bound (1.5) has been recently proved in [11] . Previously, (1.5) was proved in [7] and [9] for real-valued compactly supported L ∞ potentials. When n = 1 it was shown in [6] that we have the better bound (1.6) instead of (1.5). The method we use to prove Theorem 1.1 also allows us to get resolvent bounds when the condition (1.4) is not satisfied, which however are much weaker than the bound (1.5). More precisely, we have the following Theorem 1.2. Suppose the conditions (1.1), (1.2) and (1.3) fulfilled with δ and β satisfying either the condition
Introduction and statement of results
or the condition
Introduction and statement of results
Then, there exist constants C > 0 and h 0 > 0 independent of h and θ but depending on s and E such that for all 0 < h ≤ h 0 we have the bounds
Introduction and statement of results
Clearly, this theorem implies the following
Introduction and statement of results
Then, there exist constants C > 0 and h 0 > 0 independent of h and θ but depending on s and E such that for all 0 < h ≤ h 0 we have the bound
Introduction and statement of results
To prove the above theorems we follow the same strategy as in [11] which in turn is inspired by the paper [9] . It consists of using Carleman estimates with phase and weight functions, denoted by ϕ and µ below, depending only on the radial variable r and the parameter h, which have very weak regularity. It turns out that it suffices to choose ϕ belonging only to C 1 and µ only continuous. Thus we get derivatives ϕ ′′ and µ ′ belonging to L ∞ , which proves sufficient for the Carleman estimates to hold. Note that higher derivatives of ϕ and µ are not involved in the proof of the Carleman estimates (see the proof of Theorem 3.1 below). In order to be able to prove the Carleman estimates the functions ϕ and µ must satisfy some conditions (see the inequalities (2.2) and (2.8) below). On the other hand, to get as good resolvent bounds as possible we are looking for a phase function ϕ such that max ϕ is as small as possible. The construction of such phase and weight functions is carried out in Section 2 following that one in [11] . However, here the construction is more complicated due to the more general class the potential belongs to.
Introduction and statement of results
It is not clear if the bounds (1.5) and (1.9) are optimal for L ∞ potentials. In any case, they seem hard to improve unless one menages to construct a better phase function. By contrast, the optimality of the bound (1.6) for smooth potentials is well known (e.g. see [5] ).
The construction of the phase and weight functions revisited
We will follow closely the construction in Section 2 of [11] making some suitable modifications in order to adapte it to the more general class of potentials we consider in the present paper. We will first construct the weight function µ as follows:
The construction of the phase and weight functions revisited
, m 1 and m 2 are as in Theorem 1.2 and T 0 , T 1 , T 2 > 0 are parameters independent of h to be fixed in the proof of Lemma 2.3 below. Furthermore,
The construction of the phase and weight functions revisited
Clearly, the first derivative (in sense of distributions) of µ satisfies
The construction of the phase and weight functions revisited
The following properties of the functions µ and µ ′ are essential to prove the Carleman estimates in the next section.
The construction of the phase and weight functions revisited
Lemma 2.1. For all r > 0, r = a, we have the inequalities
The construction of the phase and weight functions revisited
Proof. It is easy to see that for r < a (2.2) follows from the inequality
The construction of the phase and weight functions revisited
Hence in this case the function f is increasing, which implies f (r) ≥ f (0) = 0 as desired. For r > a the left-hand side of (2.2) is bounded from below by
The construction of the phase and weight functions revisited
provided a is taken large enough. Furthermore, we clearly have µ ′ (r) ≥ 2k(r + 1) −1 for r < a, and hence (2.3) holds in this case, provided ǫ is taken small enough. For r > a the bound (2.3) is trivial. The bound (2.4) follows from (2.3) and the observation that µ(r) 2 ≤ (a + 1) 4k ≤ 2a 4k for all r. ✷
The construction of the phase and weight functions revisited
We now turn to the construction of the phase function ϕ ∈ C 1 ([0, +∞)) such that ϕ(0) = 0 and ϕ(r) > 0 for r > 0. We define the first derivative of ϕ by
The construction of the phase and weight functions revisited
for r > a.
The construction of the phase and weight functions revisited
Lemma 2.2. For all r ≥ 0 we have the bounds
The construction of the phase and weight functions revisited
in the other two cases. In view of the choice of ǫ, τ and a, we get the bounds
The construction of the phase and weight functions revisited
Since
The construction of the phase and weight functions revisited
Given a parameter b ≫ r 0 to be fixed in the proof of Theorem 3.1 below, independent of h, set
The construction of the phase and weight functions revisited
The following lemma will play a crucial role in the proof of the Carleman estimates in the next section.
The construction of the phase and weight functions revisited
for all r > 0, r = a.
The construction of the phase and weight functions revisited
Proof. For r < a we have
The construction of the phase and weight functions revisited
. Taking into account the definition of the parameters a and τ we conclude
The construction of the phase and weight functions revisited
for all r < a. Observe now that if (1.4) holds, we have
The construction of the phase and weight functions revisited
provided we take T 0 = 3m 0 t 0 and ǫ small enough. If (1.7) holds, we have
The construction of the phase and weight functions revisited
We take now
The construction of the phase and weight functions revisited
provided ǫ is taken small enough. On the other hand, if (1.8) holds, we have
The construction of the phase and weight functions revisited
provided we take T 2 = 6m 2 t 2 β − 1 and ǫ small enough. Using that h ǫt = e −t we conclude
The construction of the phase and weight functions revisited
Taking t 0 , t 1 and t 2 large enough, independent of h, we obtain from (2.9) and (2.10) that in all cases we have the estimate
The construction of the phase and weight functions revisited
for all r < a. We will now bound the function B from above. Note that taking h small enough we can arrange that 2b < a/2. Let first 0 < r ≤ a 2 . Since in this case we have ϕ ′ (r) ≥ Cτ (r + 1) −k with some constant C > 0, we obtain
The construction of the phase and weight functions revisited
0 is some constant depending only on b and we have used that k < (2δ − 1)/5 in all three cases. Taking h small enough, depending on τ 0 , and b big enough, independent of h and τ 0 , we get the bound
The construction of the phase and weight functions revisited
with some constant C > 0. In this case we get (2.8) from (2.11) and (2.12) by taking τ 0 big enough depending on b and C but independent of h. Let now a 2 < r < a. Then we have the bound
The construction of the phase and weight functions revisited
provided h is taken small enough. Again, this bound together with (2.11) imply (2.8) .
The construction of the phase and weight functions revisited
It remains to consider the case r > a. Using that µ = O(a 2k ) together with (2.3) we get
The construction of the phase and weight functions revisited
When (1.7) holds we have
The construction of the phase and weight functions revisited
In this case we also have
The construction of the phase and weight functions revisited
When (1.8) holds we have
The construction of the phase and weight functions revisited
We conclude from the above inequalities that
The construction of the phase and weight functions revisited
It follows from (2.13) that taking t 0 , t 1 and t 2 large enough, independent of h, we can arrange the bound
The construction of the phase and weight functions revisited
Since in this case A(r) = 0, the bound (2.14) clearly implies (2.8) . ✷
Carleman estimates
In this section we will prove the following 
Carleman estimates
with a constant C > 0 independent of h, θ and f .
Carleman estimates
Proof. We will adapt the proof of Theorem 3.1 of [11] to this more general case. We pass to the polar coordinates (r, w) ∈ R + × S n−1 , r = |x|, w = x/|x|, and recall that L 2 (R n ) = L 2 (R + × S n−1 , r n−1 drdw). In what follows we denote by · and ·, · the norm and the scalar product in L 2 (S n−1 ). We will make use of the identity
Carleman estimates
where ∆ w = ∆ w − 1 4 (n − 1)(n − 3) and ∆ w denotes the negative Laplace-Beltrami operator on S n−1 . Set u = r (n−1)/2 e ϕ/h f and
Carleman estimates
2) we can write the operator P ± (h) in the coordinates (r, w) as follows
Carleman estimates
where we have put D r = −ih∂ r and Λ w = −h 2 ∆ w . Since the function ϕ depends only on the variable r, this implies
Carleman estimates
. For r > 0, r = a, introduce the function
Carleman estimates
where V L (r, w) := V L (rw). It is easy to check that its first derivative is given by
Carleman estimates
Thus, if µ is the function defined in the previous section, we obtain the identity
Carleman estimates
Using that Λ w ≥ 0 together with (2.2) we get the inequality
Carleman estimates
In view of the assumptions (1.2) and (1.3) we have
Carleman estimates
provided b is taken large enough. Observe also that the assumption (1.1) yields
Carleman estimates
where Q b = sup |x|≤2b |V L (x)|. Combining the above inequalities we get
Carleman estimates
Now we use Lemma 2.3 to conclude that
Carleman estimates
−θh −1 µ u(r, ·) 2 + D r u(r, ·) 2 . We integrate this inequality with respect to r and use that, since µ(0) = 0, we have ∞ 0 (µ ′ F + µF ′ )dr = 0.
Carleman estimates
Thus we obtain the estimate
Carleman estimates
Using that µ = O(a 2k ) together with (2.3) and (2.4) we get from (3)
Carleman estimates
with some constant C > 0 independent of h and θ. On the other hand, we have the identity
Carleman estimates
and hence for every γ > 0. We take now γ small enough, independent of h, and recall that θ(ǫh) −1 a 2k ≤ 1. Thus, combining the estimates (3) and (3), we get 
This implies

Resolvent estimates
The bounds (1.5) and (1.9) can be derived from Theorem 3.1 in the same way as in Section 4 of [11] . Here we will sketch the proof for the sake of completeness. Observe that it follows from the estimate (3.1) and Lemma 2.2 that for 0 < h ≪ 1, 0 < θ ≤ ǫha −2k and s satisfying (2.1) we have the estimate We now combine (4.1) and (4.3) to get (4.4) (|x| + 1) −s f L 2 ≤ 4M 2 (|x| + 1) s (P (h) − E ± iθ)f L 2 .
Resolvent estimates
It follows from (4.4) that the resolvent estimate (4.5) (|x| + 1) −s (P (h) − E ± iθ) −1 (|x| + 1) −s L 2 →L 2 ≤ 4M 2 holds for all 0 < h ≪ 1, 0 < θ ≤ ǫha −2k and s satisfying (2.1). On the other hand, for θ ≥ ǫha −2k the estimate (4.5) holds in a trivial way. Indeed, in this case, since the operator P (h) is symmetric, the norm of the resolvent is upper bounded by θ −1 = O(h −2km−2 ). Finally, observe that if (4.5) holds for s satisfying (2.1), it holds for all s > 1/2 independent of h. Indeed, given an arbitrary s ′ > 1/2 independent of h, we can arrange by taking h small enough that s defined by (2.1) is less than s ′ . Therefore the bound (4.5) holds with s replaced by s ′ as desired.

{'paper_id': '119312411', 'title': 'Semi-classical resolvent estimates for short-range l $\\infty$ potentials. II', 's2_url': 'https://api.semanticscholar.org/CorpusID:119312411', 'source': 'S2ORC'}