1 Introduction and statement of results↩︎

The goal of this paper is to establish limiting absportion resolvent bounds for the semiclassical Schrödinger operator with a repulsive potential in dimensions three and higher. The dimension one case was studied in [1]. As an application, we obtain time decay of a weighted energy for the solution to the associated wave equation with a short range repulsive potential and compactly supported initial data.

To fix some notation, let \(\Delta \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\sum_{j=1}^n \partial^2_j \le 0\) be the Laplacian on \(\mathbb{R}^n\), \(n \ge 3\). We use \((r, \theta) = (|x|, x/|x|) \in (0, \infty) \times \mathbb{S}^{n-1}\) for polar coordinates on \(\mathbb{R}^n \setminus \{0\}\). Put \(B(0,r_0) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\{x \in \mathbb{R}^n : |x| < r_0\}\). For a function \(u\) defined on a subset of \(\mathbb{R}^n\), we write \(u(r, \theta) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =u(r \theta)\), and denote the radial derivative by \(u' \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\partial_ru\). If \(E \subseteq \mathbb{R}^n\) is a Borel set, \(\mathbf{1}_E\) stands for its indicator function.

Our Schrödinger operator takes the form \[\label{P} P(h) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =-h^2 \Delta + V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n),\qquad x \in \mathbb{R}^n,\tag{1}\] where \(h > 0\) is the semiclassical parameter. The conditions we place on the potential \(V : \mathbb{R}^n \to \mathbb{R}\) are as follows. We suppose \[\begin{gather} V \ge 0, \tag{2} \\ rV \mathbf{1}_{B(0,1)} \in L^\infty(\mathbb{R}^n), \tag{3} \\ V\mathbf{1}_{\mathbb{R}^n \setminus B(0,1)} \in L^\infty(\mathbb{R}^n) \tag{4}, \\ \text{for each \theta \in \mathbb{S}^{n-1}, (0, \infty) \ni r \mapsto V(r, \theta) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =V(r\theta) has bounded variation.} \tag{5} \end{gather}\]

Recall that function \(f\) of locally bounded variation on an interval \(I \subseteq \mathbb{R}\) has distributional derivative equal to a locally finite Borel measure, which we denote by \(df\); it satisfies \[\label{ftc} \int_{(a,b]} df = f^R(b) - f^R(a),\tag{6}\] for any interval \((a,b]\) contained in the interior of \(I\), where \(f^R(x) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\lim_{\delta \to 0^+} f(x + \delta)\).

The last condition we impose on \(V\) is that there exists \(C_V > 0\) so that for all \(\theta \in \mathbb{S}^{n-1}\), \[\label{V32prime32cond} dV( \cdot ,\theta) \le -C_V(r +1 )^{-1} V( \cdot ,\theta) dr.\tag{7}\] in the sense of Borel measures on \((0, \infty)\).

A prototype potential satisfying the above conditions is \[V(r, \theta) =g(\theta) \big( \mathbf{1}_{B(0,1)} r^{-1} + 2^{-1} \mathbf{1}_{\mathbb{R}^n \setminus B(0,1)} r^{-\delta} \big)\] for some \(\delta > 0\) and \(g\) a bounded function on \(\mathbb{S}^{n-1}\). Note that if \(V \in C^1(\mathbb{R}^n; [0, \infty))\), 7 implies each \(V(\cdot, \theta)\) is repulsive in sense of classical mechanics, i.e., that \(V(r, \theta) > 0\) implies \(V'(r, \theta) < 0\).

The condition 3 allows for potentials with an \(r^{-1}\)-singularity at the origin, most notably the repulsive Coulomb (\(V(x) = r^{-1}\)) and Yukawa (\(V(x) = e^{-r} r^{-1}\)) potentials [2]. Moreover, 3 and 4 imply 1 is self-adjoint with respect to the domain \(H^2(\mathbb{R}^n)\) [3]. We utilize \(H^2(\mathbb{R}^n)\) in a density argument in Appendix 6.

Our main results are the following weighted limiting absorption resolvent bounds for \(P\).

Christiansen and Datchev [1] obtained 8 and 9 for bounded, repulsive potentials on the half-line. Thus the novelty of our work is that it extends these bounds to higher dimensions for repulsive potentials which may have a singularity at \(r = 0\).

We prove Theorem 1 by means of the so-called spherical energy method, which is a popular strategy for obtaining resolvent estimates (see, e.g., [4]–[6]). It relies on separation of variables and the well known identity \[\label{conjugation321} r^{\frac{n-1}{2}}(- \Delta) r^{-\frac{n-1}{2}} = -\partial^2_r + r^{-2} \Lambda,\tag{10}\] where \[\label{Lambda} \Lambda \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =-\Delta_{\mathbb{S}^{n-1}} + \frac{(n-1)(n-3)}{4},\tag{11}\] and \(\Delta_{\mathbb{S}^{n-1}}\) denotes the negative Laplace-Beltrami operator on \(\mathbb{S}^{n-1}\). We use in a crucial way that \(\Lambda \ge 0\) on \(L^2(\mathbb{S}^{n-1})\), see 26 . This is why our approach does not cover the case \(n = 2\) where the effective potential \(-1/(4r^2)\) has a strong negative singularity as \(r \to 0\). We expect that repulsive potentials in dimension two can be treated by adapting the Mellin transform methods used in [7], [8].

As an application of 9 , we prove weighted energy decay for the solution to the wave equation \[\label{wave32eqn32srp} \begin{cases} (\partial^2_t - \Delta + V(x))u(t,x) = 0 & (t, x) \in \mathbb{R}\times \mathbb{R}^n, \, n \ge 3, \\ u(0,x) = u_0(x), \, \partial_tu(0,x) = u_1(x) & x \in \mathbb{R}^n, \end{cases}\tag{12}\] where \(u_0 \in H^1(\mathbb{R}^n)\) and \(u_1 \in L^2(\mathbb{R}^n)\) have compact support. The potential \(V\) again obeys 2 through 7 , but also the extra short range condition \[\label{stronger32bd32V} |\mathbf{1}_{\mathbb{R}^n \setminus B(0,1)} V | \le C(r + 1)^{-\delta},\tag{13}\] for some \(C > 0\) and \(\delta > 0\) such that \[\label{delta32restrictions} \delta > \begin{cases} \frac{1}{2} + \frac{n + 3}{4} & n \neq 8, \\ \frac{1}{2} + 3 & n = 8. \end{cases}\tag{14}\]

Since \(P \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =P(1) = -\Delta + V\) is self-adjoint (and nonnegative) on \(L^2(\mathbb{R}^n)\), we may use the spectral theorem for self-adjoint operators to represent the solution to 12 by \[\label{soln32spect32thm} u(t, \cdot) = \cos(t \sqrt{P}) u_0 + \frac{\sin(t \sqrt{P})}{\sqrt{P}}u_1.\tag{15}\] For \(s > 0\) fixed, define the weighted energy of the solution \(u\) to 12 to be \[E_s[u](t) = E_s(t) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\int_{\mathbb{R}^n} \langle x \rangle^{-2s}( |\partial_t u(t,x)|^2 + | \nabla u(t,x)|^2 + |u(t,x)|^2 )dx.\] Set also \[E(0) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\|\nabla u_0 \|^2_{L^2} + \| u_1 \|^2_{L^2}.\]

For smooth, nonnegative potentials of compact support, the local energy \[E_{r_0}(t) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\int_{B(0,r_0)} |\partial_t u(t,x)|^2 + | \nabla u(t,x)|^2 + |u(t,x)|^2 dx, \qquad r_0 > 0,\] obeys \[\label{compact32support32case} E_{r_0}(t) = \begin{cases} O(e^{-ct}) \text{ for some } c> 0 & n \ge 3 \text{ odd}, \\ O(t^{-2n}) & n \ge 4 \text{ even}. \end{cases}\tag{18}\] Indeed, Vainberg showed 18 for compactly supported perturbations of the Laplacian satisfying the so called Generalized Huygens Principle (defined in [9]). From a propagation of singularities result of Melrose and Sjöstrand [10], [11], the Generalized Huygens Principle holds for a large class of the so called nontrapping pertubations of Laplacian, which includes smooth, nonnegative potentials compact support. The study of energy decay for nontrapping perturbations has a long history, going back to the works of Lax, Morawetz, and Phillips [12]–[14].

On the other hand, bounds similar to 17 were obtained in previous works for various classes of short range potentials. In [15], Zappacosta considered potentials \(V \in C^1(\mathbb{R}^3; (0,\infty))\) with \(\partial^\alpha_xV = O(\langle x \rangle^{- \delta - |\alpha|})\) for all \(0 \le |\alpha| \le 1\) and some \(\delta > 2\). For the each \(\chi \in C^\infty_0(\mathbb{R}^3)\), the bound
\(\| \chi \sqrt{V}(\sin(t \sqrt{P})/\sqrt{P}) \sqrt{V} \chi \|^2_{L^2 \to L^2} = O(t^{-2})\) was obtained. Vodev extended [15] in [16] by obtaining \(E_{r_0}(t) = O(t^{-2})\) in dimension \(n \ge 3\), where \(V \in C^1(\mathbb{R}^n ; [0,\infty))\) obeys \[\begin{gather} \partial^\alpha_x V = O(\langle x \rangle^{-\delta_0}) \qquad \text{for all 0 \le |\alpha| \le 1 and some \delta_0 > (n +1)/2, and} \tag{19} \\ 2V + r \partial_rV \le C\langle x \rangle^{-\delta} \qquad \text{for some C > 0 and some \delta > 1.} \tag{20} \end{gather}\] We note that if \(V \in C^1(\mathbb{R}^n ; [0,\infty))\) satisfies 7 with \(C_V \ge 2\), then 20 holds. Vodev also obtained weighted energy decay for a class of long range, nontrapping perturbations of the Laplacian that includes perturbation by a nonnnegative long range potential, provided the initial conditions are spectrally localized away from \([0, a]\) for \(a > 0\) sufficiently large [17].

The proof of Theorem 3 proceeds as in [16], with some modifications to address the possible singular behavior of \(V\) at the origin. It suffices to establish \[\label{wave32decay32outline} t^2 E_s(t) \le C t^2 \int_t^\infty E_s(\tau) d\tau \le CE(0), \qquad t \ge 1.\tag{21}\] For this, we use Duhamel’s formula and the Fourier transform, with \(t\) dual to \(\lambda\), to represent the Fourier transform of \(u\) in terms of \((P - \lambda^2)^{-1}\), see 58 . Since the initial data are compactly supported, finite speed of propagation holds, and we may freely introduce a cutoff function \(\eta\) into this expression. So Plancherel’s theorem gives a route to control \(E_s(t)\), thanks to the bound on \(\langle x \rangle^{-s} (P - \lambda^2)^{-1}\eta\) that comes from 9 , see also Lemma 1. However, due the factor of \(t^2\) in 21 , and since multiplication by \(t\) is dual to differentiation with respect to \(\lambda\), it is necessary to control \(\langle x \rangle^{-s} \tfrac{d}{d\lambda}(P - \lambda^2)^{-1}\eta\) too. We obtain bounds on this derivative under the additional conditions 13 , 14 , and 16 , see Lemma 3.

Another energy studied is the quantity \[E^{(1)}_K[u](t) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\int_{K} |\partial_t u(t,x)|^2 + | \nabla u(t,x)|^2 + V(x)|u(t,x)|^2 dx,\] where \(K\) is suitable subset of \(\mathbb{R}^n\). In [16], Vodev considered \(K = B(0, \gamma_0 t) \subseteq \mathbb{R}^n\) for \(n \ge 3\) and suitable \(0 < \gamma_0 < 1\). He showed \(E^{(1)}_K[u](t) = O(t^{-1})\) provided 19 and 20 hold. In [18], Ikehata considered exterior subdomains \(\Omega\) of \(\mathbb{R}^n\), \(n \ge 2\), that exclude the origin. For \(K\) a compact subset of \(\Omega\), he showed \(E^{(1)}_K[u](t) = O(t^{-1})\) provided \(V\) is nonnegative, \(C^1\), and obeys \(x\cdot \nabla V + 2V \le 0\). It seems \(O(t^{-1})\)-decay was first obtained first by Morawetz [19] for \(V = 0\) in the exterior of a three-dimensional star-shaped obstacle (and later improved to exponential decay in [12]).

As alluded to above, there is a large body literature on wave decay for higher order perturbations. See [20]–[26] for more historical background.

The rest of the paper is organized as follows. In Section 2, we prove Theorem 1. In Section 3, we apply Theorem 1 to prove norm bounds for \(\langle x \rangle^{-s}(P - \lambda^2)^{-1}\langle x \rangle^{-s}\) and its \(\lambda\)-derivative. In Section 4 we prove Theorem 3. Finally, we include several appendices of technical results that assist with the proofs of earlier sections.

We thank Kiril Datchev and Georgi Vodev for helpful discussions. J. S. and A. L-H. gratefully acknowledge support from NSF DMS-2204322. J. S. was also supported by a University of Dayton Research Council Seed Grant.

2 Proof of Theorem 1↩︎

In this section, we prove Theorem 1. Throughout this section, we take \(P(h)\) as in 1 , and assume the potential \(V\) satisfies 2 through 7 .

By 10 , \[\label{conjugation322} \begin{align} P^{\pm}(h) &\mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =r^{\frac{n-1}{2}}\left( P(h) - E \pm i\varepsilon \right) r^{-\frac{n-1}{2}}\\ &= -h^2\partial^2_r + h^2r^{-2} \Lambda + V - E \pm i\varepsilon. \end{align}\tag{22}\]

Let \(u \in r^{(n-1)/2} C^\infty_0(\mathbb{R}^n)\). Define a spherical energy functional \(F[u](r)\), \[\label{F} F(r) = F[u](r) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\|hu'(r, \cdot)\|^2 - \langle (h^2 r^{-2} \Lambda + V(r, \cdot) - E)u(r, \cdot), u(r, \cdot) \rangle,\tag{23}\] where \(\| \cdot \|\) and \(\langle \cdot, \cdot \rangle\) denote the norm and inner product on \(L^2(\mathbb{S}_\theta^{n-1})\). For a weight \(w(r)\) which is absolutely continuous, nonnegative, and increasing, one computes \((wF)'\) in the sense of distributions on \((0, \infty)\): \[\label{deriv32wF} \begin{align} (wF)' &= wF' + w'F\\ &= w(-2\mathop{\rm Re}\langle (-h^2\partial^2_r + h^2r^{-2}\Lambda + V - E)u , u' \rangle \\ &+2 h^2 r^{-3} \langle \Lambda u, u \rangle - \textstyle\int_{\mathbb{S}^{n-1}} |u(\theta, r)|^2 dV(r, \theta) d\theta ) \\ &+ w' (\|hu'\|^2 - \langle h^2 r^{-2} \Lambda u, u \rangle + \langle (E - V)u, u \rangle\\ &= -2 w \mathop{\rm Re}\langle P^{\pm}(h) u, u' \rangle \mp 2\varepsilon w \mathop{\rm Im}\langle u,u'\rangle + w'\|hu'\|^2\\ &+(2wr^{-1} - w') \langle h^2r^{-2}\Lambda u,u\rangle + Ew'\| u\|^2 \\ &-\textstyle\int_{\mathbb{S}^{n-1}} |u(\theta, r)|^2 (w(r) dV(r, \theta) + w'(r)V(r, \theta)) d\theta . \end{align}\tag{24}\]

First we show 8 . Since increasing \(s\) decreases the left side of 8 , without loss of generality we may take \(0 < \delta \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =2s - 1 < 1\). We will show the last line of 24 can be made to have a suitable lower bound, using \[\label{w32general32repulsive} w(r) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =1 - \frac{C_V}{C_V + \delta}(1 + r)^{-\delta}.\tag{25}\] For such \(w\), we clearly have \[w'(r) = \frac{\delta C_V}{C_V + \delta}(r + 1)^{-1 - \delta},\] Therefore, on the one hand \[2wr^{-1} - w' = 2r^{-1}\big(1 - \frac{C_V}{C_V + \delta}(r + 1)^{-\delta} \big[ 1 - \frac{\delta}{2} \frac{r}{r + 1} \big] \big) \ge 0,\] since \(\delta < 1\). On the other hand, in the sense of measures of Borel measures on \((0, \infty)\), using 7 \[wdV + w'V = \frac{\delta C_V V}{(C_V + \delta)(r + 1)^{1 + \delta}} + wdV \le \frac{C_V V}{1 + r} ( (r +1 )^{-\delta} - 1 ) \le 0.\]

Thus, the last two estimates and 24 imply, in the sense of distributions on \((0, \infty)\),

\[\label{deriv32wF32lwr32bd} \begin{align} (wF)' &\ge -2 w \mathop{\rm Re}\langle P^{\pm}(h) u, u' \rangle \mp 2\varepsilon w \mathop{\rm Im}\langle u,u'\rangle \\ &+ w' \| hu'\|^2 + E w'\| u\|^2. \end{align}\tag{26}\] Integrating 26 with respect to \(dr\) from \(r = r_0 > 0\) to \(r = \infty\) implies (because \(u\) is compactly supported) \[\label{pre32est32after32integrating} \int_{r_0}^\infty E w' \| u\|^2 + w' \| hu'\|^2 dr + w(r_0) F(r_0) \le \int_{r_0}^\infty 2 w \mathop{\rm Re}\langle P^{\pm}(h) u, u' \rangle \pm 2\varepsilon w \mathop{\rm Im}\langle u,u'\rangle dr.\tag{27}\]

Since \(u = r^{(n-1)/2} v\) for some \(v \in C^\infty_0(\mathbb{R}^n)\), we recognize that \[\label{Fr0} \begin{align} F(r_0) &= \|hu'(r_0, \cdot) \|^2 + r_0^{n-3} \langle h^2 \Delta_{\mathbb{S}^{n-1}}v(r_0, \cdot), v(r_0, \cdot) \rangle \\ &+(Er^{n-1}_0 - h^24^{-1}(n-1)(n-3)r_0^{n-3}) \|v(r_0, \cdot)\|^2 + r_0^{n-1} \langle V(r_0, \cdot)v(r_0, \cdot), v(r_0, \cdot) \rangle. \end{align}\tag{28}\] We rewrite the term in 28 involving \(\Delta_{\mathbb{S}^{n-1}}\) using the well known formula for the Laplacian in spherical coordinates: \[r^{-2}\Delta_{\mathbb{S}^{n-1}} = \Delta - \partial^2_r - (n-1)r^{-1} \partial_r.\] Therefore, \[\label{rewrite32laplace32beltrami} \begin{align} r_0^{n-3}& \langle h^2 \Delta_{\mathbb{S}^{n-1}}v(r_0, \cdot), v(r_0, \cdot) \rangle\\ &= h^2 r_0^{n-1}\langle (\Delta v)(r_0, \cdot), v(r_0, \cdot) \rangle \\ &- h^2r_0^{n-1}\langle (\partial^2_rv)(r_0, \cdot), v(r_0, \cdot) \rangle - h^2(n-1)r_0^{n-2}\langle (\partial_rv)(r_0, \cdot), v(r_0, \cdot) \rangle. \end{align}\tag{29}\] We can express the differential operators \(\partial_r\) and \(\partial^2_r\) with respect to the Euclidean coordinate system, \[\label{r32derivatives} \partial_r = r^{-1} \sum_{j=1}^n x_j \partial_{x_j}, \qquad \partial^2_r = r^{-2} \sum_{k=1}^n x_k \sum_{j=1}^n x_j \partial_{x_k}\partial_{x_j}.\tag{30}\] Thus by 29 and 30 , all terms in 28 tend to zero as \(r_0 \to 0\), except for possibly \(\|hu'(r, \cdot) \|^2\) in dimension three, which in that case tends to \(|v(0)|^2\int_{\mathbb{S}^{n-1}} d\theta\). We conclude \[\lim_{r_0 \to 0} w(r_0) F(r_0) = w(0) F(0) = \begin{cases} \omega_{n-1} w(0)|v(0)|^2 & n = 3, \\ 0 & n \ge 4,\end{cases}\] where \(\omega_{n-1}\) is the \((n-1)\)-dimensional volume of \(\mathbb{S}^{n-1}\).

Thus in view of 27 and \(0 < w \le 1\), \[\label{est32after32integrating} \begin{align} \int_{0}^\infty Ew' \| u\|^2 &+ w' \| hu'\|^2 dr \\ &\le 2 \big( \int_{0}^\infty \frac{1}{h^2w'} \| P^{\pm}(h) u \|^2 dr \big)^{1/2} \big(\int_{0}^\infty w'\| hu'\|^2 dr \big)^{1/2} \\ &+ \frac{2\varepsilon}{h} (\int_0^\infty \|u\|^2 dr)^{1/2} (\int_0^\infty \|hu'\|^2 dr)^{1/2}. \end{align}\tag{31}\] We now estimate, \[\begin{align} \int_0^\infty \|hu'\|^2dr &= \mathop{\rm Re}\int_0^\infty \langle u, -h^2u'' \rangle dr \\ &= \mathop{\rm Re}\big( \int_{0}^\infty \langle u, P^\pm(h)u \rangle dr + \int_0^\infty \langle u, (E - V - h^2 r^{-2}\Lambda)u \rangle dr \mp i \varepsilon\int^\infty_0 \|u\|^2 dr \big)\\ &= \mathop{\rm Re}\int_{0}^\infty \langle u, P^\pm(h)u \rangle dr + \int_0^\infty \langle u, (E - V - h^2 r^{-2}\Lambda)u \rangle dr \\ &\le \big( \int_0^\infty \frac{1}{w'} \|P^\pm(h)u\|^2dr \big)^{1/2} \big(\int_0^\infty w'\|u\|^2dr \big)^{1/2} + E\int_0^\infty \|u\|^2dr, \end{align}\] and \[\begin{align} \varepsilon\int_0^\infty \|u\|^2 dr &= \varepsilon\| v\|^2_{L^2} \\ &= |\mathop{\rm Im}\langle (P(h) - E \pm i\varepsilon)v, v \rangle_{L^2}| \\ &= \big| \mathop{\rm Im}\int_0^\infty \langle P^{\pm}(h)u, u \rangle dr \big| \\ &\le \big( \int_0^\infty \frac{1}{w'} \|P^\pm(h)u\|^2dr \big)^{1/2} \big(\int_0^\infty w'\|u\|^2dr \big)^{1/2}. \end{align}\] Combining these gives

\[\frac{\varepsilon^2}{h^2} \int_0^\infty \|u\|^2 dr \cdot \int_0^\infty \|hu'\|^2dr \le (E + \varepsilon) \int_0^\infty \frac{1}{h^2w'} \|P^\pm(h)u\|^2dr \cdot \int_0^\infty w'\|u\|^2dr.\] Plugging this into 31 yields

\[\label{est32after32integrating322} \begin{align} \int_{0}^\infty Ew' \| u\|^2 + w'\| hu'\|^2 dr &\le 2 \big( \int_0^\infty \frac{1}{h^2w'} \|P^\pm(h)u\|^2dr \big)^{1/2}\\ &\cdot \big( \big(\int_0^\infty w'\|hu'\|^2dr \big)^{1/2} + (E + \varepsilon)^{1/2} \big(\int_0^\infty w'\|u\|^2dr \big)^{1/2} \big). \end{align}\tag{32}\]

Completing the square in 32 , we find

\[\label{complete32the32square} \begin{align} \big( E^{1/2} \big( \int_0^\infty& w'\|u\|^2dr \big)^{1/2} - \frac{(E+\varepsilon)^{1/2}}{E^{1/2}} \big( \int_0^\infty \frac{1}{h^2w'} \|P^\pm(h)u\|^2dr \big)^{1/2} \big)^2 \\ &+ \big( \big( \int_0^\infty w'\|hu'\|^2dr \big)^{1/2} - \big( \int_0^\infty \frac{1}{h^2w'} \|P^\pm(h)u\|^2dr \big)^{1/2} \big)^2\\ &\le \frac{2E + \varepsilon}{E} \int_0^\infty \frac{1}{h^2w'} \|P^\pm(h)u\|^2dr. \end{align}\tag{33}\] Dropping the second term on the left side of 33 implies \[\label{E32and32ep32est} E^{1/2} \big( \int_0^\infty w'\|u\|^2dr \big)^{1/2} \le \big(\frac{(E+\varepsilon)^{1/2}}{E^{1/2}} + \frac{(2E + \varepsilon)^{1/2}}{E^{1/2}} \big) \big( \int_0^\infty \frac{1}{h^2w'} \|P^\pm(h)u\|^2dr \big)^{1/2}.\tag{34}\]

Next, consider the sector \(\{ z \in \mathbb{C}: |\mathop{\rm Im}z| < \alpha \mathop{\rm Re}z \}\) for \(0 < \alpha < 1\). Since
\(w' = C_V \delta (C_V + \delta)^{-1} (r + 1)^{-1-\delta}\) and \(\delta = 2s -1\), from 34 , we get for all \(h > 0\),
\(E \pm i \varepsilon\in \{ z \in \mathbb{C}: |\mathop{\rm Im}z| < \alpha \mathop{\rm Re}z \}\), and \(u \in r^{n-1/2}C^\infty_0(\mathbb{R}^n)\),

\[\label{alpha32est} \begin{align} |(E &\pm i \varepsilon)^{1/2}|\big( \int_0^\infty (r +1)^{-2s}\|u\|^2dr \big)^{1/2} \\ &\le h^{-1} (1 + \alpha^2)^{1/4} \big( \frac{1}{\delta} + \frac{1}{C_V} \big)((1 + \alpha)^{1/2} + (2 + \alpha)^{1/2}) \big( \int_0^\infty (r +1)^{2s} \|P^\pm(h)u\|^2dr \big)^{1/2}, \end{align}\tag{35}\] Here, our branch of the complex square root is chosen so that \(\mathop{\rm Im}(E \pm i\varepsilon)^{1/2} > 0\), and we used that \(|(E \pm i\varepsilon)|^{1/2} = (E^2 + \varepsilon^2)^{1/4} \le E^{1/2}(1 + \alpha^2)^{1/4}\) for \(E \pm i \varepsilon\in \{ z \in \mathbb{C}: |\mathop{\rm Im}z| < \alpha \mathop{\rm Re}z \}\).

Since \(u \in r^{(n-1)/2} C^\infty_0(\mathbb{R}^n)\), a standard density argument, which we review in Appendix 6, shows that 35 implies \[\label{resolv32est32in32sector} \begin{align} \|z^{1/2}(r + 1)^{-s}(P(h) &-z)^{-1}(r +1)^{-s}\|_{L^2 \to L^2} \\ &\le h^{-1}(1 + \alpha^2)^{1/4} \big( \frac{1}{\delta} + \frac{1}{C_V} \big)((1 + \alpha)^{1/2} + (2 + \alpha)^{1/2}), \end{align}\tag{36}\] on \(\{ z \in \mathbb{C}: |\mathop{\rm Im}z| < \alpha \mathop{\rm Re}z \}\) and for any \(0 < \alpha < 1\). To extend this bound to all \(z \in \mathbb{C}\setminus [0, \infty)\), we use the Phragmén-Lindelöf principle [27] in the following way. For \(u, v \in L^2(\mathbb{R}^n)\), put \[U(z) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =z^{1/2} \langle (r + 1)^{-s}(P(h) -z)^{-1}(r +1)^{-s}u, v \rangle_{L^2}.\] Then \(U(z)\) in analytic in \(\Omega_\alpha \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\{ z \in \mathbb{C}: \alpha \mathop{\rm Re}z < |\mathop{\rm Im}z| \}\). By 36 , on \(\partial \Omega_\alpha \setminus \{0 \}\) we have \[\label{U32on32bdry} |U(z)| \le h^{-1}(1 + \alpha^2)^{1/4} \big( \frac{1}{\delta} + \frac{1}{C_V} \big) ((1 + \alpha)^{1/2} + (2 + \alpha)^{1/2}) \|u\|_{L^2} \|v\|_{L^2}.\tag{37}\] On the other hand, in \(\Omega_\alpha\), we have the standard bound \[\label{std32bd} |U(z)| \le \frac{|z|^{1/2}\|u \|_{L^2} \|v\|_{L^2}}{\mathop{\rm dist}(z,[0, \infty))} = \begin{cases} \frac{\|u \|_{L^2} \|v\|_{L^2}}{|z|^{1/2}} & \mathop{\rm Re}z < 0, \\ \frac{|z|^{1/2}\|u \|_{L^2} \|v\|_{L^2}}{|\mathop{\rm Im}z|} & \mathop{\rm Re}z \ge 0, \, z \in \Omega_\alpha, \end{cases}\tag{38}\] where we used \[\frac{1}{\mathop{\rm dist}(z,[0, \infty))} = \frac{1}{ \inf_{r \ge 0} ((\mathop{\rm Re}z - r)^2 + (\mathop{\rm Im}z)^2)^{1/2}} = \begin{cases} \frac{1}{|z|} & \mathop{\rm Re}z <0 , \\ \frac{1}{|\mathop{\rm Im}z|} & \mathop{\rm Re}z \ge 0, \, z \in \Omega_\alpha. \end{cases}\]

Finally, define, \(g(z) = e^{i (z^{-1})^{1/2}}\), where our branch of the square root is as above. In \(\Omega_\alpha\), \(|g(z)| \le e^{-c|z|^{-1/2}}\) for some \(0 < c < 1\) depending on \(\alpha\). This with 38 says that for any \(\sigma > 0\), \[\label{sigma32bd} \limsup_{z \to 0, \, z \in \Omega_\alpha} |g(z)|^{\sigma} |U(z)| = 0.\tag{39}\] Therefore, from 37 and 39 , the Phragmén Lindelöf Theorem (Theorem 6 in Appendix 5) implies that 36 holds for all \(z \in \Omega_\alpha\) too. Sending \(\alpha \to 0^+\) completes the proof of 8 .

To prove 9 , start again at 33 and drop the first term on the left hand side. Still working on \(\{ z \in \mathbb{C}: |\mathop{\rm Im}z| < \alpha \mathop{\rm Re}z \}\), some manipulations give \[\label{uprimebound} \left(\int_0^\infty w'\|hu'\|^2\,dr\right)^{1/2}\leq (1+\sqrt{2+\alpha})\left(\int_0^\infty \frac{1}{h^2 w'} \|P^{\pm}(h)u\|^2 dr\right)^{1/2}.\tag{40}\] By integration by parts, \[\begin{align} \int_0^\infty (r +1)^{-3-\delta}\|u\|^2\,dr&=\frac{2}{2+\delta}\int_0^\infty (r +1)^{-2-\delta}\mathop{\rm Re}\langle u,u'\rangle\,dr\\ &\leq h^{-1}\left(\int_0^\infty(r +1)^{-1-\delta}\|hu'\|^2\,dr\right)^{1/2}\left(\int_0^\infty (r +1)^{-3-\delta}\|u\|^2\,dr\right)^{1/2}, \end{align}\] which implies \[\label{poincare} \left(\int_0^\infty (r + 1)^{-3-\delta}\|u\|^2\,dr\right)^{1/2}\leq h^{-1}\left(\int_0^\infty(r +1)^{-1-\delta}\|hu'\|^2\,dr\right)^{1/2}.\tag{41}\]

From 40 , 41 and \(w' = C_V \delta (C_V + \delta)^{-1} (r + 1)^{-1-\delta}\), \[\left(\int_0^\infty (r+1)^{-3-\delta}\|u\|^2\,dr\right)^{1/2}\leq h^{-2}\big( \frac{1}{\delta} + \frac{1}{C_V} \big) (1+\sqrt{2+\alpha}) \left(\int_0^\infty (r +1)^{1 + \delta}\|P^{\pm}(h)u\|^2\,dr\right)^{1/2}.\] Using again the density argument in Appendix 6, for \(0 < \delta < 1\), \[\label{resolv32est32in32sector32low32freq} \|(1+r)^{-\frac{3+\delta}{2}}(P(h)-z)^{-1}(1+r)^{-\frac{1+\delta}{2}} \|_{L^2 \to L^2 }\leq h^{-2}\left(\delta^{-1}+C_V^{-1}\right) (1+\sqrt{2+\alpha}),\tag{42}\] on \(\{ z \in \mathbb{C}: |\mathop{\rm Im}z| < \alpha \mathop{\rm Re}z \}\). Then, as above, 38 , the Phragmén Lindelöf Theorem, and sending \(\alpha \to 0^+\), imply \[\|(1+r)^{-\frac{3+\delta}{2}}(P(h)-z)^{-1}(1+r)^{-\frac{1+\delta}{2}} \|_{L^2 \to L^2 }\leq h^{-2}\left(\delta^{-1}+C_V^{-1}\right) (1+\sqrt{2}), \qquad z \in \mathbb{C}\setminus [0, \infty).\] Since the norm of an operator and its adjoint coincide, \[\|(1+r)^{-\frac{1+\delta}{2}} (P(h)-z)^{-1} (1+r)^{-\frac{3+\delta}{2}} \|_{L^2 \to L^2 }\leq h^{-2}\left(\delta^{-1}+C_V^{-1}\right) (1+\sqrt{2}), \qquad z \in \mathbb{C}\setminus [0, \infty).\] The three lines lemma then says that for fixed \(z \in \mathbb{C}\setminus [0, \infty)\), the analytic mapping \[\lambda \mapsto (1+r)^{-\frac{3+\delta}{2} +\lambda }(P(h)-z)^{-1}(1+r)^{-\frac{1+\delta}{2} - \lambda}, \qquad 0 < \mathop{\rm Re}\lambda < 1,\] (with values in the space of bounded operators \(L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)\)) obeys \[\label{rest} \| (1+r)^{-\frac{3+\delta}{2} + \theta }(P(h)-z)^{-1}(1+r)^{-\frac{1+\delta}{2} - \theta} \|_{L^2 \to L^2} \le h^{-2}\left(\delta^{-1}+C_V^{-1}\right) (1+\sqrt{2}), \qquad \theta \in [0, 1].\tag{43}\]

Having established 43 , to finish, we need to see that we can choose \(\delta\) and \(\theta\) appropriately to arrive at 9 . That is, we need to attain the more general weights characterized by \(s_1, s_2 > 1/2\), \(s_1 + s_2 > 2\). However, because we have the restrictions \(\delta \in (0,1)\) and \(\theta \in [0,1]\), we first need to make reductions as follows. Since decreasing \(s_1\) or \(s_2\) in 9 increases the right side, it suffices to suppose \(s_1, \, s_2 > 1/2\), \(2< s_1 + s_2 < 3\). Furthermore, by taking the adjoint, it is no restriction to have \(s_1 \le s_2\). If we write \(s_1 = (1 + 2\delta_1)/2\) for some \(\delta_1 > 0\), then we may replace \(s_2\) by \(\min(s_2, (3 + \delta_1)/2 )\). Having made these reductions, 9 follows from 43 by setting \(\delta = s_1 + s_2 -2 < 1\), \(\theta = (s_2 - s_1 + 1)/2 \le (4 - \delta_1)/4 < 1\).

3 Resolvent bounds for wave decay↩︎

In this section, we consider the operator \(P = P(1) = -\Delta + V\), with \(P(h)\) as in 1 ; \(V\) obeys 2 through 7 . As a consequence of Theorem 1, we prove several more resolvent bounds for \(P\), which enable us in Section 4 to establish weighted energy decay for the solution to the wave equation 12 . Throughout this section, \(C\) denotes a positive constant whose precise value may change, but is always independent \(\lambda\), which plays the role of our spectral parameter.

Next, we prove an estimate for the derivative of the weighted resolvent, which requires an extra short range condition on the potential. As input we need the following bound for the weighted square of the free resolvent, which we prove in Appendix [deriv free resolv appendix].

4 Proof of Theorem 3↩︎

In this section we prove Theorem 3 by combining the resolvent bounds of the previous section with an argument appearing in [16]. As before we use the notation \(P = -\Delta + V : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)\), \(n \ge 3\), where \(V\) obeys 2 through 7 along with 13 and 14 .

In several steps below, we use that for all \(0 \le \alpha \le 1\), there exists \(C > 0\) so that for any \(f \in H^1(\mathbb{R}^n)\), \[\label{Poincare32etc} \| V^\alpha f \|^2_{L^2} \le C (\| \nabla f \|^2_{L^2} + \|f\|^2_{L^2}) \le C\| \nabla f \|^2_{L^2}.\tag{54}\] The first inequality follows from 3 , 13 and Lemma 7, while the second follows from the Poincaré inequality (as we work in dimension \(n \ge 3\)).

Given \(s > 0\) and \(u\) as in 15 solving the wave equation 12 , with compactly supported initial conditions \(u(0, x) = u_0(x) \in H^1(\mathbb{R}^n)\), \(\partial_t u(0,x) = u_1(x) \in L^2(\mathbb{R}^n)\), define \[\begin{gather} E_s(t) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\int_{\mathbb{R}^n} \langle x \rangle^{-2s}( |\partial_t u(t,x)|^2 + |\nabla u(t,x)|^2 + |u(t,x)|^2)dx, \\ E(0) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\| \nabla u_0 \|^2_{L^2} + \| u\|^2_{L^2}. \end{gather}\]

The local energy decay 17 follows from 56 and

5 Phragmén-Lindelöf Theorem↩︎

In this appendix we recall the Phragmén Lindelöf Theorem. Let \(f(z)\) be a holomorphic function in a domain \(D\) of the complex plane with boundary \(\Gamma\). We say that \(f(z)\) does not exceed a number \(M \ge 0\) in modulus at a boundary point \(\zeta \in \Gamma\) if \(\limsup_{z \to \zeta, \, z \in D} |f(z)| \le M\).

6 Density argument: proof of 36 and 42↩︎

In this appendix, we prove 36 and 42 as a consequence of

7 Justification of Remark 2↩︎

In the setting of Theorem 1, consider the case of \(V = 0\) and \(n =3\). In that scenario the integral kernel of \((P(h) - z)^{-1} = (-h^2 \Delta -z)^{-1}\) with \(z \in \mathbb{C}\setminus [0, \infty)\) is given by \[R_0(x,y,z) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =h^{-2} \frac{e^{i \tfrac{\sqrt{z}}{h}|x -y|}}{4\pi |x -y|}, \qquad \mathop{\rm Im}\sqrt{z} > 0.\] We recall why having a bound like 8 on \(\langle \cdot \rangle^{-s_1}(-h^2 \Delta -z)^{-1}\langle \cdot \rangle^{-s_2}: L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3)\) requires \(s_1, \, s_2 > 1/2\).

Since the norm of an operator and its adjoint coincide, it suffices to show \(s_1 > 1/2\) is necessary. Use \(\sqrt{z}\) of the form \(\sqrt{z} = E + i\varepsilon\) for \(E > 0\) fixed and \(\varepsilon> 0\) tending to zero. Then, as calculated in the proof of [30], for \(f \in C^\infty_0(\mathbb{R}^3)\), \[\label{outgoing32asymptotic} \begin{align} \langle x \rangle^{-s_1} \int_{\mathbb{R}^3} &R_0(x,y,z) f(y) dy \\ &= h^{-2} \frac{\langle x \rangle^{-s_1}}{4\pi |x|} e^{\frac{i}{h}(E+ i\varepsilon)|x|} (\hat{f} \big( \tfrac{E}{h} \tfrac{x}{|x|} \big) + o(1)) + O(|x|^{-2}), \quad \text{as \varepsilon\to 0^+ and |x| \to \infty.} \end{align}\tag{69}\] If \(f\) is chosen so that \(|\hat{f}| > c\) for some \(c > 0\) on \(\{|x| = E/h\}\), 69 and \(s_1 \le 1/2\) imply
\(\| \langle x \rangle^{-s_1} \textstyle\int_{\mathbb{R}^3} R_0(x,y,z) f(y) dy \|_{L^2} \to \infty\) as \(\varepsilon\to 0^+\).

Next, supposing \(s_1, \, s_2 > 1/2\), we show why a bound like 9 on \(\langle \cdot \rangle^{-s_1}(-h^2 \Delta -z)^{-1}\langle \cdot \rangle^{-s_2}: L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3)\) requires additionally that \(s_1 + s_2 \ge 2\), which is nearly the condition we impose for 9 . Using \(\sqrt{z} = i \varepsilon\) for \(\varepsilon> 0\) tending to zero, and \(f_\eta(y) = \langle y \rangle^{-\eta- \frac{3}{2}}\), \(\eta > 0\), we see as in [32] that \[\begin{align} \langle x \rangle^{-s_1} \int_{\mathbb{R}^3} &R_0(x,y,z) \langle y \rangle^{-s_2} f(y) \\ &= h^{-2} \langle x \rangle^{-s_1} \int_{\mathbb{R}^3} \frac{e^{-\frac{\varepsilon}{h}|x-y|}}{4\pi |x- y|} \langle y \rangle^{-s_2} f(y) dy \\ &\gtrsim h^{-2} e^{-\frac{3\varepsilon}{2h} |x|} \langle x \rangle^{-s_1-1} \int_{|y| \le \frac{|x|}{2}} \langle y \rangle^{-s_2- \eta - \frac{3}{2}} dy \gtrsim h^{-2} e^{-\frac{3\varepsilon}{2h} |x|} \langle x \rangle^{-s_1 - s_2- \eta + \frac{1}{2}}, \end{align}\] where the implicit constants indicated by \(\gtrsim\) are independent of \(\varepsilon\) and \(\eta\). First sending \(\varepsilon\to 0^+\) gives \(s_1 + s_2 \ge 2 -\eta\), but since \(\eta > 0\) is arbitrary, we in turn get \(s_1 + s_2 \ge 2\).

To see that the \(O(|z|^{-\frac{1}{2}} h^{-1})\)-dependence of the right side of 8 is optimal, consider the function \(u = e^{i \frac{\sqrt{z}}{h} x_1} \chi\) for nontrivial \(\chi \in C^\infty_0(\mathbb{R}^3 ; [0, 1])\). We have \[\langle x \rangle^{s}(-h^2 \Delta -z)u = -i \sqrt{z} h \langle x \rangle^{s} \partial_{x_1} \chi - h^2 e^{i \frac{\sqrt{z}}{h}x_1} \langle x \rangle^s \Delta \chi =\mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}}f,\] whence \(\langle \cdot \rangle^{-s} (-h^2 \Delta - z)^{-1} \langle \cdot \rangle^{-s} f = \langle \cdot \rangle^{-s}u\) and thus, as \(h \to 0\), \[\frac{\|\langle \cdot \rangle^{-s} (-h^2 \Delta - z)^{-1} \langle \cdot \rangle^{-s} f\|_{L^2}}{\|f \|_{L^2} } = \frac{\| \langle \cdot \rangle^{-s} u\|_{L^2}}{\|f \|_{L^2} } \gtrsim |z|^{-\frac{1}{2}} h^{-1}.\]

Finally, we argue why the \(O(h^{-2})\)-dependence of the right side of 9 is sharp. As noted before [29] gives that \(R_{0, s_1, s_2}(\lambda) = \langle \cdot \rangle^{-s_1}(-h^2 \Delta -\lambda^2)^{-1}\langle \cdot \rangle^{-s_2}\) (\(s_1,\, s_2 > 1/2\), \(s_1 + s_2 > 2\)), has extends continuously from \(\mathop{\rm Im}\lambda > 0\) to \(\mathbb{R}\) in the space of bounded opeartors \(\mathbb{R}^3 \to \mathbb{R}^3\). In this case, we have, \[h^{-2}\big\| \langle x \rangle^{-s_1} \int_{\mathbb{R}^3} \frac{1}{4\pi |x -y|}\langle y \rangle^{-s_2}dy \big \|_{L^2 \to L^2} = \lim_{\varepsilon\to 0} \| R_{0, s_1, s_2}(i\varepsilon)\|_{L^2 \to L^2}.\]

8 Proof of Lemma 2↩︎

In this appendix we prove Lemma 2. The proof proceeds in two steps. First we treat the case \(|\lambda| \ge 1\), followed by \(|\lambda| \le 1\).

9 Useful lemmas↩︎

References↩︎