1 Introduction↩︎

Let \(\mathcal{O}\subseteq\mathbb{R}^d\), \(d\ge 2\), be a possibly empty, bounded domain with smooth boundary such that \(\Omega=\mathbb{R}^d\setminus\mathcal{O}\) is connected. In this paper we investigate the magnetic Schrödinger operator \[\label{eq:1461} P=(i\nabla+b(x))^2+V(x) : L^2(\Omega) \to L^2(\Omega)\tag{1}\] from the viewpoint of resolvent estimates. The magnetic potential \(b : {\mathbb{R}}^d \to {\mathbb{R}}^d\) and electric potential \(V : {\mathbb{R}}^d \to {\mathbb{R}}\) are assumed to have \(L^\infty\) regularity. Leveraging these resolvent estimates, our primary goal is to establish conditions for exponential weighted energy decay for solutions to the associated wave equation \[\label{eq:1462} \begin{cases} (\partial^2_t + P)u(t,x) = 0 & \text{in } {\mathbb{R}}\times \Omega, \\ u(t,x) = 0 & \text{on } {\mathbb{R}}\times \partial \Omega, \\ u(0,x) = f_1(x), \, \partial_tu(0,x) = f_2(x) & \text{in } \Omega. \end{cases}\tag{2}\]

To set the stage, let us recall the classical results for the free wave equation, where both \(b\) and \(V\) are identically zero. In this simpler setting, if \(\mathcal{O} = \emptyset\) (so \(\Omega = {\mathbb{R}}^d\)), Huygen’s principle implies that when \(d \ge 3\) is odd, the energy of the solution to 2 within any fixed compact set decays to zero in finite time. On the other hand, when \(\Omega \neq \emptyset\) (i.e, \(\Omega\) is an exterior domain, with \(b\) and \(V\) still vanishing), the decay of local energy for solutions to 2 is related to the dynamics of the underlying Hamiltonian flow. The non-trapping condition, where all geodesics escape to infinity, is well known to be related to rapid energy decay. This condition is known to hold for specific geometries, such as convex obstacles and more generally for obstacles where an escape function can be constructed. Foundational works by Lax, Morawetz, Phillips, Ralston and Strauss [1]–[4] address such scenarios and the resulting decay, utilizing multiplier methods and associated properties of the resolvent of the Laplacian. More broadly, for non-trapping geometries, the resolvent satisfies a characteristic high frequency bound [5]. Our assumption for obstacles, stated as 4 below, is a resolvent estimate of this type.

When non-zero potentials \(b\) and \(V\) are present, results on energy decay draw from the work of Vainberg [6] and Melrose-Sjöstrand [7], [8]. For example, in the case of smooth potentials of compact support, the local energy is known to decay like \(O(e^{-Ct})\) for some \(C > 0\) when \(d \ge 3\) is odd, and like \(O(t^{-2d})\) when \(d \ge 2\) is even. Vainberg [6] showed these decay rates apply to compactly supported perturbations of the Laplacian satisfying the Generalized Huygens Principle (as defined in [9]). Work by Melrose and Sjöstrand on the propagation of singularities [7], [8] further implies that this principle is satisfied by a broad class of smooth non-trapping perturbations of the Laplacian, including those with smooth compactly supported potentials.

We now specify the assumptions for our analysis. In what follows, \(\|\cdot\|\) and \(\|\cdot\|_1\) denote the operator norms \(L^2(\Omega)\to L^2(\Omega)\) and \(H^1(\Omega)\to L^2(\Omega)\), respectively. We consider two main scenarios:
a) \(\mathcal{O} = \emptyset\) (so that \(\Omega = {\mathbb{R}}^d\)), \(d\ge 3\),
b) \(\mathcal{O} \neq \emptyset\) and \(b\equiv 0\).
For both cases we assume the potentials satisfy: \[\label{eq:1463} |V(x)|+|b(x)|\le Ce^{-c\langle x\rangle},\tag{3}\] where \(\langle x \rangle \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =(|x|^2+1)^{1/2}\) and \(C,c>0\) are some constants. In case b) (exterior domain, \(V\) only), we impose a non-trapping condition on the obstacle \(\mathcal{O}\) via a high frequency resolvent estimate for the Dirichlet Laplacian \(\widetilde{P} =-\Delta\) on \(L^2(\Omega)\). Let \(\chi\in C^\infty(\mathbb{R}^d; [0,1])\) be of compact support such that \(\chi =1\) near \(\overline{\mathcal{O}}\). Define multiplication by \(\chi\) on \(L^2(\Omega)\) by \(u \mapsto \chi \rvert_\Omega u\). The operator \(\widetilde{P}\) can be viewed as a black box Hamiltonian in the sense of Sjöstrand and Zworski [10], as defined in [5]. By the analytic Fredholm theorem [5], the cutoff resolvent \(\chi (\widetilde{P} - \lambda^2)^{-1} \chi: L^2(\Omega) \to D(\widetilde{P})\) continues meromorphically from \(\{\text{Im} \lambda < 0\}\) to the whole complex plane \(\mathbb{C}\) if \(d\) is odd, and to the Riemann surface of the logarithm if \(d\) is even. The poles of this continuation are called resonances. Our non-trapping condition for case b) is the high-frequency bound: \[\label{eq:1464} \left\|\chi(\widetilde{P}-\lambda^2)^{-1}\chi\right\|\le C\lambda^{-1},\quad \lambda \ge\lambda_0,\tag{4}\] for some constants \(C,\lambda_0>0\). We note that the norm in 4 is bounded by a constant also for \(0\le\lambda \le\lambda_0\) for arbitrary obstacles, as proved in [11]. No separate non-trapping condition is imposed for case a) beyond the condition 3 .

Hereafter, \(P\) denotes the self-adjoint realization of the operator \((i\nabla+b)^2+V\) on the Hilbert space \(L^2(\mathbb{R}^d)\) in the case a), and the Dirichlet self-adjoint realization of \(-\Delta+V\) in the case b). For the case a), Appendix 8 details the construction of \(P\) via a quadratic form on \(H^1({\mathbb{R}}^d)\). For case b), we recall from [12] ]that the domain of \(P\) is the intersection \(H^1_0(\Omega) \cap H^2(\Omega)\) of Sobolev spaces (we define \(H^1_0(\Omega)\) as the closure in \(H^1\)-norm of smooth and compactly supported functions on \(\Omega\)). In case b) we will at times make use of the Green formula \[\langle \nabla u, \nabla v \rangle_{L^2(\Omega)} = \langle u, -\Delta v\rangle_{L^2(\Omega)}, \qquad u, \, v \in H^1_0(\Omega) \cap H^2(\Omega).\]

In both scenarios, we assume \(P\ge 0\), for which a sufficient condition is \(V\ge 0\). The domain of the square root of a nonnegative self-adjoint operator coincides with its quadratic form domain [13]. Consequently, in case a), the form domain is \(H^1({\mathbb{R}}^d)\), while in case b) it is \(H^1_0(\Omega)\).

The solution to the wave equation (2 ) is given by \[\label{eq:1465} u(t)=\cos\left(t\sqrt{P}\right)f_1+P^{-1/2}\sin\left(t\sqrt{P}\right)f_2.\tag{5}\] We define the weight \(\mu(x)=e^{-c\langle x\rangle/2}\). For some of our results it is important to suppose that zero is neither an eigenvalue nor a resonance of \(P\). More precisely, we require the following low frequency resolvent bound: there exist constants \(C> 0\) and \(\delta_0 \ll 1\) such that \[\label{eq:1466} \sum_{\ell=0}^1\left\|\mu\nabla^\ell(P-\lambda^2\pm i\varepsilon)^{-1}\mu\right\|\le C,\qquad 0<\lambda\le\delta_0,\tag{6}\] holds uniformly in \(0<\varepsilon\le 1\). The condition 6 is established in Section 5 in dimensions \(d\ge 5\), under suitable short range conditions on \(b\) and \(V\), and provided \(V \ge 0\) (see 58 ). Our main result is then:

This paper approaches the proof of Theorem 1 by employing resolvent estimates derived from Carleman estimates and various perturbation arguments. This strategy has precedents in related areas, including low-frequency resolvent estimates or expansions for blackbox, short-range, or nontrapping perturbations [14]–[19], high-frequency resolvent bounds for the magnetic Schrödinger operator [20]–[22], and Strichartz and smoothing estimates for the magnetic Schrödinger operator [23], [24]. The novelty of our method lies in its suitability for handling \(L^\infty\) coefficients and its flexibility across a range of frequencies, essential for achieving the decay described in Theorem 1. In Section 2, we establish Carleman estimates for the free Laplacian on \({\mathbb{R}}^d\), applicable to both medium and high frequencies (Propositions 4 and 5). The Carleman estimates facilitate the derivation of limiting absorption resolvent bounds at medium and high frequencies for cases a) and b), which are Theorems 6 and 9, respectively. For these resolvent bounds, it is enough to suppose short range conditions weaker than the exponential decay. For case b), we employ a resolvent remainder argument, see 41 and 42 , to transfer the high-frequency nontrapping bound 4 to the perturbed resolvent. The smallness of the remainder at high frequency, captured by 42 , does not apply in the case of a first order perturbation. That is why we assume \(b\) vanishes when \(\mathcal{O} \neq \emptyset\).

Subsequently, in Sections 3 and 4, under the assumption 3 , we utilize resolvent identities to extend these limiting absorption bounds to the meromorphic continuation of the weighted resolvent \(\mu(P -\lambda^2)^{-1}\mu\) (Theorems 8 and 10). These identities allow us to leverage the meromorphic continuation of the free resolvent (see Appendix 10 for a review of its properties). This is the step where the exponential decay of the coefficients plays a key role. Finally, Section 6 demonstrates how these resolvent estimates lead to the exponential decay rates presented in ?? and ?? . Furthermore, we show \(\psi_{\delta, t} \equiv 1\) can be chosen when the dimension is odd and condition 6 is met. The arguments in this section draw inspiration from [25], which established polynomial-in-time decay for wave equation solutions on unbounded Riemannian manifolds with general smooth, nontrapping metrics. However, modifications to these arguments are introduced in our setting to exploit the meromorphic continuation of the resolvent, enabling us to obtain exponential decay.

Future directions: We do not treat the case \(\mathcal{O}=\emptyset\) and \(d=2\) in the presence of a magnetic potential. The obstruction comes from the Carleman estimate in Proposition 4, which in two dimensions has a loss near the origin. This loss is related to the effective potential arising after separation of variables in polar coordinates, which has a negative singularity at the origin when \(d =2\). In case b), we avoid this by using the local Carleman estimate of [26]; however, that estimate applies to electric potentials only and for Dirichlet boundary condition. We note that when \(d=2\) and \(\mathcal{O}=\emptyset\), the necessary Carleman estimate is known to hold for sufficiently large \(\lambda\) [21].

We anticipate that in even dimensions, under condition 6 , it is also possible take \(\psi_{\delta, t} \equiv 1\) in ?? and ?? , provided the right-hand sides are modified to \(Ct^{-2d}\). To achieve this it seems necessary to demonstrate control on the derivatives of the resolvent all the way down to zero frequency, as we have in odd dimensions (see Theorems 8 and 10).

It would also be interesting to investigate time decay for short-range \(L^\infty\) potentials–those not necessarily exhibiting exponential decay. In such scenarios, the weighted resolvent is unlikely to possess a meromorphic continuation. Consequently, control over the derivatives of the resolvent would require different arguments.

We thank Kiril Datchev for helpful discussions. J. S. and A. L-H. gratefully acknowledge support from NSF DMS-2204322.

2 Carleman estimates for the Euclidean Laplacian↩︎

Let \(r=|x|\) be the radial variable and define a Lipschitz function \(\omega\) by \[\omega(r)= \left\{ \begin{array}{lll} (r+1)^{2\ell}&for& 0\le r\le A,\\ (A+1)^{2\ell}\left(1+(A+1)^{-2s+1}-(r+1)^{-2s+1}\right)&for& r\ge A, \end{array} \right.\] with parameters \(A\gg 1\) and \(s,\ell\) satisfying \[0<s-\frac{1}{2}<\ell<\frac{2s}{3}<\frac{2}{3}.\] Its first derivative is given by \[\omega'(r)= \left\{ \begin{array}{lll} 2\ell(r+1)^{2\ell-1}&for& 0\le r<A,\\ (2s-1)(A+1)^{2\ell}(r+1)^{-2s}&for& r>A. \end{array} \right.\] We also define a function \(\varphi\in C^1([0,+\infty))\) such that \(\varphi(0)=0\) and its first derivative is a Lipschitz function given by \[\varphi'(r)= \left\{ \begin{array}{lll} (r+1)^{-\ell}(2-(r+1)^{-\kappa})&for& 0\le r\le A,\\ K_A(r+1)^{-2s}&for& r\ge A, \end{array} \right.\] where \[0<\kappa<2s-1,\quad \kappa<1-\ell,\] and \[K_A=(A+1)^{2s-\ell}(2-(A+1)^{-\kappa})=O(A^{2s-\ell}).\] The main properties of the functions \(\omega\) and \(\varphi\) are given in the next lemma.

Set \[P_{0,\varphi}(\tau)=-e^{\tau\varphi}\Delta e^{-\tau\varphi},\] where \(\tau\gg 1\) is a large parameter. Given a parameter \(0<h\le 1\) we will denote by \(H^1_h({\mathbb{R}}^d)\) the Sobolev space \(H^1(\mathbb{R}^d)\) equipped with the norm \(\|\cdot\|_{H^1_h}\) defined by \[\|f\|_{H^1_h}^2 \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\|f\|_{L^2}^2+h^{2}\|\nabla f\|_{L^2}^2.\] Furthermore, \(H^{-1}_h\) will denote the dual space of \(H^1_h\) with respect to the scalar product \(\langle\cdot,\cdot\rangle_{L^2}\) with the norm \[\|f\|_{H^{-1}_h} \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\sup_{0\neq g\in H^1_h}\frac{|\langle f,g\rangle_{L^2}|}{\|g\|_{H^1_h}}.\] Let the function \(\psi\in C_0^\infty(\mathbb{R}^d)\) be such that \(\psi(x)=1\) for \(|x|\le 1\). We first prove the following

Proof. We will write \(P_{0,\varphi}(\tau)\) in the polar coordinates \((r,w)\in\mathbb{R}^+\times\mathbb{S}^{d-1}\), \(r=|x|\), \(w=x/|x|\). Recall that \(L^2(\mathbb{R}^d)=L^2(\mathbb{R}^+\times\mathbb{S}^{d-1}, r^{d-1}drdw)\). In what follows we denote by \(\|\cdot\|_0\) and \(\langle\cdot,\cdot\rangle_0\) the norm and the scalar product in \(L^2(\mathbb{S}^{d-1})\). We take complex conjugation to occur in the first argument of \(\langle\cdot,\cdot\rangle_0\). We make use of the identity \[\label{eq:2467} r^{(d-1)/2}\Delta r^{-(d-1)/2}=\partial_r^2+r^{-2}\Delta_w-(d-1)(d-3)(2r)^{-2},\tag{7}\] where \(\Delta_w\) denotes the negative Laplace-Beltrami operator on \(\mathbb{S}^{d-1}\). Using (7 ), we write the operator \[\mathcal{P}_\varphi(\tau)=r^{(d-1)/2}(P_{0,\varphi}(\tau)-\lambda^2)r^{-(d-1)/2}\] in the form \[{\mathcal{P}}_\varphi(\tau)=D_r^2+r^{-2}(\Lambda+c_d)-\lambda^2-2i\tau\varphi'D_r+V_1+V_2,\] where \(D_r=i\partial_r\), \(\Lambda=-\Delta_w\), and \[\begin{align} V_1&=-\tau^{2}(\varphi')^2, &&V_2=(d-1)(d-3)(2r)^{-2}+\tau\varphi'', &&c_d=0, &\text{ if } d=2, \\ V_1&=-\tau^{2}(\varphi')^2, &&V_2=\tau\varphi'', &&c_d=(d-1)(d-3)/4, &\text{ if } d\ge 3. \end{align}\] Set \(u(r,w)=r^{(d-1)/2}(1-\psi(rw))f(rw)\) and, for \(r>0\), \(r\neq A\), \[E(r)=-\left\langle (r^{-2}(\Lambda+c_d) -\lambda^2+V_1)u(r,\cdot),u(r,\cdot)\right\rangle_0+\|D_ru(r,\cdot)\|_0^2.\] For the first derivative of \(E\), we get in the sense of distributions on \((0, \infty)\), \[\begin{align} E'(r) &=\frac{2}{r}\left\langle r^{-2}(\Lambda+c_d)u,u\right\rangle_0-\left\langle V_1'u,u\right\rangle_0 +4\tau\varphi'\|D_ru\|_0^2\\ &-2{\rm Im}\,\left\langle\mathcal{P}_\varphi(\tau)u,D_ru\right\rangle_0 + 2{\rm Im}\,\left\langle V_2u,D_ru\right\rangle_0. \end{align}\] If \(\omega\) is as above, we have the identity \[\begin{align} (\omega E)' &=\omega'E+\omega E' \\ &= (2r^{-1}\omega-\omega')\left\langle r^{-2}(\Lambda+c_d)u,u\right\rangle_0+\left\langle(\lambda^2\omega'-(\omega V_1)')u,u\right\rangle_0 \\ &+(\omega'+4\tau\varphi'\omega)\|D_ru\|_0^2 +2\omega{\rm Im}\,\left\langle V_2u,D_ru\right\rangle_0 \\ &-2\omega{\rm Im}\,\left\langle(\mathcal{P}_\varphi(\tau)\pm i\varepsilon)u,D_ru\right\rangle_0 \mp 2\varepsilon\omega{\rm Im}\,\left\langle u,D_ru\right\rangle_0. \end{align}\] For \(x \in {\rm supp}\, u\) and \(r = |x|\), we have \[\label{eq:2468} |V_2(r)|\lesssim \begin{cases} \tau(r+1)^{-1-\ell} &for\quad r<A,\\ \tau A^{2s-\ell}(r+1)^{-1-2s}+(r+1)^{-2} &for\quad r>A. \end{cases}\tag{8}\] In what follows \(C>0\) will be a constant which may depend on \(\delta\) but is independent of \(h\) and \(\lambda\). Its precise value may change from line to line. We have the lower bound \[\begin{align} (\omega E)'(r) &\ge (2r^{-1}\omega-\omega')\left\langle r^{-2}(\Lambda + c_d) u,u\right\rangle_0+(\lambda^2\omega'-(\omega V_1)')\left\|u\right\|_0^2 \\ &+(\omega'/2+\tau\omega\varphi')\|D_ru\|_0^2-C\omega^2|V_2|^2(\omega'+\tau\omega\varphi')^{-1}\left\|u\right\|_0^2 \\ &-C\omega^2(\omega'+\tau\omega\varphi')^{-1}\left\|(\mathcal{P}_\varphi(\tau)\pm i\varepsilon)u\right\|_0^2 -2\varepsilon\omega\left\|u\|_0\|D_ru\right\|_0 \\ &\ge(2r^{-1}\omega-\omega')\left\langle r^{-2}\Lambda u,u\right\rangle_0+n(r)\left\|u\right\|_0^2+\tau\omega\varphi'\|D_ru\|_0^2 \\ &-C\tau^{-1}\omega(\varphi')^{-1}\left\|(\mathcal{P}_\varphi(\tau)\pm i\varepsilon)u\right\|_0^2 -2\varepsilon\omega\left\|u\|_0\|D_ru\right\|_0, \end{align}\] where \[\begin{align} n(r) &=\lambda^2\omega'-(\omega V_1)'-C\omega^2|V_2|^2(\tau\omega\varphi')^{-1} \\ &= \lambda^2\omega'+\tau^{2}(\omega(\varphi')^2)'-C\tau^{-1}|V_2|^2\omega(\varphi')^{-1}. \end{align}\] When \(r < A\), \(\omega (\varphi')^{-1}\lesssim (r + 1)^{3\ell}\). Thus in view of ?? and 8 , since \(0 < \kappa < 2s-1\), \(\kappa < 1 - \ell\), we have for \(r<A\), \[\begin{align} n(r) &\ge \lambda^2\omega'+2\kappa\tau^{2}(r+1)^{-1-\kappa}-C\tau(r+1)^{-2+\ell} \\ &= \lambda^2\omega'+\kappa\tau (r+1)^{-1-\kappa} ( 2\tau - C\kappa^{-1}(r+1)^{\kappa - (1- \ell)}) \\ &\ge\lambda^2\omega'+\kappa\tau^{2}(r+1)^{-1-\kappa}\\ &\ge\lambda^2\omega'+\kappa \tau^{2}(r+1)^{-2s}, \end{align}\] provided \(\tau\) is large enough. To bound \(n(r)\) from below for \(r>A\) observe that, in view of (8 ), in this case we have the bounds \[\begin{align} \frac{|V_2(r)|^2\omega(r)}{\omega'(r)\varphi'(r)}& \lesssim A^{\ell-2s}(r+1)^{4s}|V_2(r)|^2 \\ &\lesssim \tau^2A^{2s-\ell}(r+1)^{-2}+ A^{\ell-2s}(r+1)^{4s-4} \\ &\lesssim \tau^2A^{2s-2-\ell}+ A^{\ell+2s-4}\\ &\lesssim \tau^2A^{2s-1-2\ell}. \end{align}\] To get the last inequality we used \(2s - 2 - \ell, \, \ell + 2s -4 < 2s - 1 - 2\ell\). From this and (?? ), for \(r>A\), we get \[\begin{align} n(r) &\ge \omega'\left(\lambda^2-C\tau^{2}A^{2s-1-2\ell}\right) \\ &=\omega'\left(\lambda^2-CA_0^{2s-1-2\ell}\right)\\ &\ge 2\lambda^2\omega'/3, \end{align}\] provided \(A_0\) is taken large enough. Combining this with \(\lambda \ge \delta\) and \[\omega'(r) \ge (2s -1) A_0 \tau^{4\ell/(1 + 2\ell -2s)} (r + 1)^{-2s} \ge (2s -1) A_0 \tau^{2} (r + 1)^{-2s}\] for \(r > A\), we get, taking \(A_0\) larger if necessary, \[n(r) \ge \lambda^2\omega'/2+\tau^2(r+1)^{-2s}, \qquad r > A.\] From the above inequalities, \[\begin{align} (\omega E)'(r)&\ge(2r^{-1}\omega-\omega')\left\langle r^{-2}(\Lambda +c_d) u,u\right\rangle_0+2^{-1}(\omega'\lambda^2+\kappa(r+1)^{-2s}\tau^2)\left\|u\right\|_0^2\\ &+\tau\omega\varphi'\|D_ru\|_0^2-C\tau^{-1}\omega(\varphi')^{-1}\left\|(\mathcal{P}_\varphi(\tau)\pm i\varepsilon)u\right\|_0^2 -2\varepsilon\omega\left\|u\|_0\|D_ru\right\|_0. \end{align}\] Integrating this inequality and using that \[\int_0^\infty (\omega E)'(r)dr=0,\] we obtain \[\label{eq:2469} \begin{align} &\int_0^\infty(2r^{-1}\omega-\omega')\left\langle r^{-2}(\Lambda u + c_d),u\right\rangle_0dr+\int_0^\infty(\omega'\lambda^2 +(r+1)^{-2s}\tau^2) \left\|u\right\|_0^2dr\\ &+\tau\int_0^\infty \omega\varphi'\|D_ru\|_0^2dr\\ &\lesssim \tau^{-1}\int_0^\infty \omega(\varphi')^{-1}\left\|(\mathcal{P}_\varphi(\tau)\pm i\varepsilon)u\right\|_0^2dr +\varepsilon\int_0^\infty\omega\|u\|_0\|D_ru\|_0dr. \end{align}\tag{9}\] Observe now that \[\omega(r)\varphi'(r)^{-1}\lesssim (r+1)^{2s},\quad \omega(r)\lesssim A^{2\ell}.\] In view of Lemma 1 we also have \[\omega'(r)\gtrsim (r+1)^{-2s},\quad \omega(r)\varphi'(r)\gtrsim (r+1)^{-2s},\quad 2r^{-1}\omega(r)-\omega'(r)\gtrsim (r+1)^{-2s}.\] Therefore (9 ) implies the estimate \[\label{eq:24610} \begin{align} &\int_0^\infty(r+1)^{-2s} \langle r^{-2}(\Lambda + c_d)u, u \rangle_0dr+(\lambda^2+\tau^2)\int_0^\infty(r+1)^{-2s}\|u\|_0^2dr\\ &+\tau\int_0^\infty(r+1)^{-2s}\|D_ru\|_0^2dr\\ &\lesssim \tau^{-1}\int_0^\infty(r+1)^{2s}\left\|(\mathcal{P}_\varphi(\tau)\pm i\varepsilon)u\right\|_0^2dr+A^{2\ell}\varepsilon\int_0^\infty\left(\gamma\|D_ru\|_0^2+\gamma^{-1}\|u\|_0^2\right)dr, \end{align}\tag{10}\] for every \(\gamma>0\). On the other hand, in view of the identity \[{\rm Re}\,\int_0^\infty\langle 2i\varphi' D_ru,u\rangle_0 dr=\int_0^\infty \varphi''\|u\|_0^2dr,\] we obtain \[\begin{align} {\rm Re}\,\int_0^\infty\langle(\mathcal{P}_\varphi(\tau)\pm i\varepsilon)u,u\rangle_0 dr &=\int_0^\infty\|D_ru\|_0^2dr +\int_0^\infty \langle r^{-2}(\Lambda+c_d)u,u\rangle_0 dr\\ &-\int_0^\infty(\lambda^2+\tau^{2}\varphi'^2+\widetilde{c}_dr^{-2})\|u\|_0^2dr\\ &\ge \int_0^\infty\|D_ru\|_0^2dr-O(\lambda^2+\tau^{2})\int_0^\infty\|u\|_0^2dr, \end{align}\] where \(\widetilde{c}_d=1/4\) if \(d=2\) and \(\widetilde{c}_d=0\) if \(d\ge 3\). This implies \[\label{eq:24611} \int_0^\infty\|D_ru\|_0^2dr\lesssim (\lambda^2+\tau^{2})\int_0^\infty\|u\|_0^2dr +\tau^{-2}\int_0^\infty\|(\mathcal{P}_\varphi(\tau)\pm i\varepsilon)u\|_0^2dr.\tag{11}\] By (10 ) with \(\gamma=(\lambda+\tau)^{-1}\) and (11 ), \[\label{eq:24612} \begin{align} &\int_0^\infty(r+1)^{-2s}\langle r^{-2}(\Lambda + c_d)u, u \rangle_0dr+(\lambda+\tau)^2\int_0^\infty(r+1)^{-2s}\left\|u\right\|_0^2dr\\ &+\int_0^\infty(r+1)^{-2s}\|D_ru\|_0^2dr\\ &\lesssim\tau^{-1}\int_0^\infty(r+1)^{2s}\left\|(\mathcal{P}_\varphi(\tau)\pm i\varepsilon)u\right\|_0^2dr +A^{2\ell}\varepsilon(\lambda+\tau)\int_0^\infty\|u\|_0^2dr. \end{align}\tag{12}\] We will now show that (12 ) implies (?? ). Since \[r^{-2}( \Lambda + c_d) = -\Delta + r^{-(d-1)/2} \partial^2_r r^{(d-1)/2} - \tilde{c}_d r^{-2},\] for any \(0<\epsilon\ll 1\) independent of \(\tau\) and \(\lambda\), \[\begin{align} \epsilon^2 \int_0^\infty &(r+1)^{-2s} \langle r^{-2}(\Lambda + c_d) u, u \rangle_0 dr \\ &= \epsilon^2 \int_0^\infty(r+1)^{-2s} r^{d-1} \langle r^{-2}(\Lambda + c_d) (1 - \psi) f, (1 - \psi) f \rangle_0 dr \\ &\ge \epsilon^2 \int_0^\infty(r+1)^{-2s} r^{d-1} \langle -\Delta (1 - \psi) f, (1 - \psi) f\rangle_0 dr + \epsilon^2 \int_0^\infty(r+1)^{-2s} \langle \partial^2_r u, u \rangle_0 dr \\ & \gtrsim \epsilon^2 \langle -\Delta(1 -\psi) f, (1- \psi) \langle x \rangle^{-2s} f \rangle_{L^2} + \epsilon^2 \int_0^\infty(r+1)^{-2s} \langle \partial^2_r u, u \rangle_0 dr \\ & \gtrsim \epsilon^2 \| \langle x \rangle^{-s}\nabla(1 -\psi) f \|^2_{L^2} - O(\epsilon^2) \| \langle x \rangle^{-s} (1 - \psi) f \|^2_{L^2} + \epsilon^2 \int_0^\infty(r+1)^{-2s} \langle \partial^2_r u, u \rangle_0 dr. \end{align}\] Now integrate by parts \[\label{eq:24613} \begin{align} \int_0^\infty(r+1)^{-2s} \langle \partial^2_r u, u \rangle_0 dr = &-\int_0^\infty(r+1)^{-2s} \|D_r u \|^2_0 dr \\ &+\int_0^\infty 2s (r+1)^{-2s-1} \| u \|^2_0 dr. \end{align}\tag{13}\] Therefore (12 ) implies \[\label{eq:24614} \begin{align} &(\lambda+\tau)\|\langle x\rangle^{-s}(1-\psi)f\|_{L^2} +\epsilon\|\langle x\rangle^{-s}\nabla((1-\psi)f)\|_{L^2}\\ &\lesssim \epsilon\|\langle x\rangle^{-s}(1-\psi)f\|_{L^2}+\tau^{-1/2}\|\langle x\rangle^{s} (P_{0,\varphi}(\tau)-\lambda^2\pm i\varepsilon)(1-\psi)f\|_{L^2}\\ &+A^{\ell}\varepsilon^{1/2}(\lambda+\tau)^{1/2}\|f\|_{L^2}. \end{align}\tag{14}\] If \(\epsilon\ll\delta\) we can absorb the first term in the right-hand side of (14 ) by the first term in the left-hand side and obtain (?? ).

Finally, we explain why (?? ) holds when \(d\ge 3\). That is, when \(d \ge 3\), we may take \(\psi \equiv 0\) and thus \(u(r,w)=r^{(d-1)/2}f(rw)\). In this case, 8 holds for all \(x\) with \(r = |x|\neq A\). From this, one shows we again have 12 . The subsequent estimates follow as before. In particular, if \(d \ge 3\), the Poincaré inequality (Lemma 10) ensures convergence of the right side of 13 .

 \(\Box\)

In what follows we improve the estimate ?? with \(\psi\equiv 0\) when \(d\ge 3\), showing that it still holds with the norm \(\|\cdot\|_{L^2}\) in the first term in the right-hand side replaced by the smaller Sobolev norm \(\|\cdot\|_{H_h^{-1}}\), where still \(h = (\lambda + \tau)^{-1}\) and \(\lambda \ge \delta\), \(\tau \ge \tau_0\) are as in the statement of Proposition 4. To this end we again utilize the operator \(P_{0, \varphi}(\tau) = -e^{\tau \varphi} \Delta e^{-\tau \varphi}\) and its generalization \[P_{0,\varphi_p}(\tau) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\langle x\rangle^{p}P_{0,\varphi}(\tau)\langle x\rangle^{-p} =-\Delta+\mathcal{Q}_p, \quad \mathcal{Q}_p=2\tau\nabla\varphi_p\cdot\nabla-\tau^{2}|\nabla\varphi_p|^2+\tau\Delta\varphi_p, \qquad p \in {\mathbb{R}},\] where \[\varphi_p(r)=\varphi(r)+\frac{p}{2}\tau^{-1}\log(r^2+1).\] By integration by parts \[\begin{gather} \langle \mathcal{Q}_p f, g \rangle_{L^2} = \langle f, \mathcal{Q}^*_p g \rangle_{L^2}, \qquad f,g \in H^1({\mathbb{R}}^d), \\ \mathcal{Q}^*_p \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =-2\tau\nabla\varphi_p\cdot\nabla-\tau^{2}|\nabla\varphi_p|^2-\tau\Delta\varphi_p. \end{gather}\]

It is easy to see that \[|\nabla\varphi_p|\lesssim |\varphi'(r)|+\tau^{-1}\lesssim 1,\] where the constants implicit in the estimate depend on \(p\) but are independent of \(\tau\). Furthermore, since \[\Delta\varphi_p=\varphi_p''(r)+\frac{d-1}{r}\varphi'_p(r),\quad r\neq 0, \, A,\] we also have \[|\Delta\varphi_p|\lesssim 1+r^{-1},\quad r \neq 0,\,A.\] Therefore, by Poincaré’s inequality, ?? , \(P_{0, \varphi_p}(\tau)\) maps boundedly \(H^2({\mathbb{R}}^d) \to L^2({\mathbb{R}}^d)\). Additionally, \[\begin{align} \|h^{2}\mathcal{Q}_pf\|_{L^2}&\lesssim (h\tau+h^{2}\tau^{2})\|f\|_{H_h^1}+h^2\tau\|r^{-1}f\|_{L^2}\\ &\lesssim (h\tau+h^{2}\tau^{2})\|f\|_{H_h^1}+h^2\tau\|\nabla f\|_{L^2}\\ &\lesssim (h\tau+h^{2}\tau^{2})\|f\|_{H_h^1}, \qquad f \in H^1({\mathbb{R}}^d), \end{align}\] and similarly for \(\mathcal{Q}^*_p\), where we have again used Poincaré’s inequality. Hence \[\label{eq:24615} \|h^{2}\mathcal{Q}_p\|_{H_h^1\to L^2} \lesssim h\tau+h^{2}\tau^{2}\lesssim 1.\tag{15}\]

Proof. Recall that \(\|f\|_{H_h^{s}}\sim\|(1-h^{2}\Delta)^{s/2}f\|_{L^2}\), \(s=-1,1\). Using this it is easy to see that the above bounds hold for \(p=0\) and \(P_{0,\varphi}(\tau)\) replaced by \(-\Delta\).

To prove ?? through ?? , begin by using 15 in combination with ?? in the case \(p =0\) and \(P_{0, \varphi}(\tau)\) replaced by \(-\Delta\). We get that for \(\theta \gg 1\), \[\| h^2 \mathcal{Q}_p (-h^2 \Delta \pm i \theta^2)^{-1}\|_{L^2 \to L^2} \le \| h^2 \mathcal{Q}_p\|_{H^1_h \to L^2} \| (-h^2 \Delta \pm i \theta^2)^{-1}\|_{L^2 \to H^1_h} \lesssim \theta^{-1} \le 1/2,\] whence \(I + h^2 \mathcal{Q}_p(-h^2 \Delta \pm i \theta^2)^{-1}\) is invertible \(L^2({\mathbb{R}}^d) \to L^2({\mathbb{R}}^d)\) by a Neumann series. It is then checked by direct computation that the inverse of \[h^2 P_{0, \varphi_p}( \tau) \pm i \theta^2 = -h^2 \Delta + h^2 \mathcal{Q}_p \pm i\theta^2 : H^2({\mathbb{R}}^d) \to L^2({\mathbb{R}}^d)\] is \[\label{eq:24620} (-h^2 \Delta + h^2 \mathcal{Q}_p \pm i\theta^2)^{-1} = (-h^2 \Delta \pm i \theta^2)^{-1} (I + h^2 \mathcal{Q}_p (-h^2 \Delta \pm i \theta^2)^{-1})^{-1}, \qquad \theta \gg 1.\tag{16}\] From this we also conclude the identity \[\label{eq:24621} \begin{align} &\left(-h^{2}\Delta+h^{2}\mathcal{Q}_p\pm i\theta^2\right)^{-1}- \left(-h^{2}\Delta\pm i\theta^2\right)^{-1}\\ &=\left(-h^{2}\Delta\pm i\theta^2\right)^{-1}h^2\mathcal{Q}_p \left(-h^{2}\Delta+h^{2}\mathcal{Q}_p\pm i\theta^2\right)^{-1}. \end{align}\tag{17}\] Using this strategy we can also establish bounded invertibility of \(-h^2 \Delta + h^2 \mathcal{Q}^*_p \pm i\theta^2 : H^2({\mathbb{R}}^d) \to L^2({\mathbb{R}}^d)\) for \(\theta\) large.

We further show that for \(\theta\) big enough \[\label{eq:24622} \langle x\rangle^{p}\left(h^{2}P_{0,\varphi}(\tau)\pm i\theta^2\right)^{-1}\langle x\rangle^{-p} = (-h^2 \Delta + h^2 \mathcal{Q}_p \pm i \theta^2)^{-1}, \qquad p \in {\mathbb{R}}.\tag{18}\] where initially the left side is interpreted as an operator sending \(C^\infty_0({\mathbb{R}}^d)\) to \(H^2_{\text{loc}}({\mathbb{R}}^d)\). At first, let \(p \ge 0\). Suppose \(f, \, g \in C^\infty_0({\mathbb{R}}^d)\); choose a sequence \(u_k \in C^\infty_0({\mathbb{R}}^d)\) converging to \((-h^2 \Delta + h^2 \mathcal{Q}_{p} \pm i \theta^2)^{-1}f\) in the \(H^2({\mathbb{R}}^d)\)-norm. Then \[\begin{align} \int_{{\mathbb{R}}^d} &\overline{g} \left(h^{2}P_{0,\varphi}(\tau)\pm i\theta^2\right)^{-1}\langle x\rangle^{-p} f \\ & = \int_{{\mathbb{R}}^d} \overline{g} \left(-h^2 \Delta + h^2 \mathcal{Q}_0 \pm i\theta^2\right)^{-1}\langle x\rangle^{-p} f \\ &= \lim_{k \to \infty} \int_{{\mathbb{R}}^d} \left[ \left(-h^2 \Delta + h^2 \mathcal{Q}^*_0 \mp i\theta^2\right)^{-1} \overline{g}\right]\langle x\rangle^{-p} (-h^2 \Delta + h^2 \mathcal{Q}_{p} \pm i \theta^2)u_k \\ &= \lim_{k \to \infty} \int_{{\mathbb{R}}^d} \left[ \left(-h^2 \Delta + h^2 \mathcal{Q}^*_0 \mp i\theta^2\right)^{-1} \overline{g} \right] (-h^2 \Delta + h^2 \mathcal{Q}_0 \pm i \theta^2) \langle x\rangle^{-p} u_k \\ & = \int_{{\mathbb{R}}^d} \overline{g} \langle x\rangle^{-p} (-h^2 \Delta + h^2 \mathcal{Q}_p \pm i \theta^2)^{-1}f, \end{align}\] This confirms 18 for \(p \ge 0\). To see 18 for \(p < 0\), we give a calculation analogous to the previous one but replace \(P_{0,\varphi}(\tau)\) with its adjoint \(-h^2 \Delta + h^2 \mathcal{Q}^*_0\). This yields
\(\langle x\rangle^{p}\left(-h^2 \Delta + h^2 \mathcal{Q}^*_0 \pm i\theta^2\right)^{-1}\langle x\rangle^{-p} = (-h^2 \Delta + h^2 \mathcal{Q}^*_p \pm i \theta^2)^{-1}\) for \(p \ge 0\), which implies 18 for \(p < 0\) by duality.

Now we are in a position to show ?? . By 15 , 17 , and since 18 holds for any \(p \in {\mathbb{R}}\): \[\begin{align} & \left \| \langle x\rangle^{p}\left(h^{2}P_{0,\varphi}(\tau)\pm i\theta^2\right)^{-1}\langle x\rangle^{p}\right\|_{H_h^{-1}\to H_h^1} = \left\|\left(-h^2 \Delta + h^2 \mathcal{Q}_p \pm i \theta^2 \right)^{-1} \right\|_{H_h^{-1}\to H_h^1}\\ &\le \left\|\left(-h^{2} \Delta\pm i\theta^2\right)^{-1}\right\|_{H_h^{-1}\to H_h^1}\\ & +\left\|\left(-h^{2}\Delta\pm i\theta^2\right)^{-1} \right\|_{L^2\to H_h^1}\|h^{2}\mathcal{Q}_p\|_{H_h^1\to L^2} \left\|\left(-h^2 \Delta + h^2 \mathcal{Q}_p \pm i \theta^2 \right)^{-1} \right\|_{H_h^{-1}\to H_h^1}\\ &\lesssim 1+\theta^{-1}\left\|\left(-h^2 \Delta + h^2 \mathcal{Q}_p \pm i \theta^2 \right)^{-1} \right\|_{H_h^{-1}\to H_h^1}, \end{align}\] which implies ?? if \(\theta\) is taken large enough; ?? can be obtained in the same way. On the other hand we obtain ?? and ?? from 16 and 18 .  \(\Box\)

We derive from Proposition 4 and Lemma 2 the following

Proof. We use the identity \[\begin{align} f&=h^{2}\left(\mp i(\varepsilon+(\theta/h)^2)+\lambda^2\right)\left(h^{2}P_{0,\varphi}(\tau)\mp i\theta^2\right)^{-1}f\\ &+h^{2}\left(h^{2}P_{0,\varphi}(\tau)\mp i\theta^2\right)^{-1}(P_{0,\varphi}(\tau)-\lambda^2\pm i\varepsilon)f. \end{align}\] Set \[g=\left(h^{2}P_{0,\varphi}(\tau)\mp i\theta^2\right)^{-1}f.\] By Lemma 2, for \(\theta\) large enough, \[\begin{align} \|\langle x\rangle^{-s}f\|_{L^2}\lesssim & \left\|\langle x\rangle^{-s}g\right\|_{L^2}+h^2\left\|\left(h^{2}P_{0,\varphi}(\tau)\mp i\theta^2\right)^{-1}\right\|_{H^{-1}_h\to L^2} \left\|(P_{0,\varphi}(\tau)-\lambda^2\pm i\varepsilon)f\right\|_{H^{-1}_h}\\ & \lesssim \left\|\langle x\rangle^{-s}g\right\|_{L^2}+h^2 \left\|(P_{0,\varphi}(\tau)-\lambda^2\pm i\varepsilon)f\right\|_{H^{-1}_h}. \end{align}\] Here and later in the proof the implicit constants depend on \(\theta\) but are independent of \(\lambda\) and \(\tau\). We now apply (?? ) to the function \(g\). Note that \(g\) satisfies the required hypothesis of Proposition 4 because by Lemma 2 \[\begin{align} \langle x \rangle^s (&P_{0, \varphi}(\tau) - \lambda^2 \pm i \varepsilon)g \\ &=\langle x \rangle^s (P_{0, \varphi}(\tau) - \lambda^2 \pm i \varepsilon) \left(h^{2}P_{0,\varphi}(\tau)\mp i\theta^2\right)^{-1}f \\ &=\left(\langle x \rangle^s \left(h^{2}P_{0,\varphi}(\tau)\mp i\theta^2\right)^{-1} \langle x \rangle^{-s} \right) \langle x \rangle^{s} (P_{0, \varphi}(\tau) - \lambda^2 \pm i \varepsilon) f \in L^2({\mathbb{R}}^d). \end{align}\] Therefore, combining ?? with Lemma 2, \[\begin{align} &\|\langle x\rangle^{-s}g\|_{H^1_h}\lesssim h\tau^{-1/2}\|\langle x\rangle^{s} (P_{0,\varphi}(\tau)-\lambda^2\pm i\varepsilon)g\|_{L^2}+A^{\ell}(\varepsilon h)^{1/2}\|g\|_{L^2}\\ &\lesssim h\tau^{-1/2}\left\|\langle x\rangle^{s}\left(h^{2}P_{0,\varphi}(\tau)\mp i\theta^2\right)^{-1} \langle x\rangle^{-s}\right\|_{H^{-1}_h\to L^2} \|\langle x\rangle^{s}(P_{0,\varphi}(\tau)-\lambda^2\pm i\varepsilon)f\|_{H_h^{-1}}\\ &+A^{\ell}(\varepsilon h)^{1/2} \left\|\left(h^{2}P_{0,\varphi}(\tau)\mp i\theta^2\right)^{-1}\right\|_{L^2\to L^2}\|f\|_{L^2}\\ &\lesssim h\tau^{-1/2} \|\langle x\rangle^{s}(P_{0,\varphi}(\tau)-\lambda^2\pm i\varepsilon)f\|_{H_h^{-1}}+A^{\ell}(\varepsilon h)^{1/2}\|f\|_{L^2}. \end{align}\] Thus we obtain \[\|\langle x\rangle^{-s}f\|_{H^1_h}\lesssim h\left(h+\tau^{-1/2}\right) \|\langle x\rangle^{s}(P_{0,\varphi}(\tau)-\lambda^2\pm i\varepsilon)f\|_{H_h^{-1}}+A^{\ell}(\varepsilon h)^{1/2}\|f\|_{L^2},\] which implies ?? since \(h<\tau^{-1}\).  \(\Box\)

3 Resolvent bounds for the magnetic Schrödinger operator↩︎

Consider in \(\mathbb{R}^d\), \(d\ge 3\), the operator \[P=(i\nabla+b(x))^2+V(x),\] where the electric potential \(V\in L^\infty(\mathbb{R}^d,\mathbb{R})\) and the magnetic potential \(b\in L^\infty(\mathbb{R}^d,\mathbb{R}^d)\) satisfy \[\label{eq:3461} |V(x)|+|b(x)|\le C\langle x\rangle^{-\rho}, \quad C>0,\,\rho>1.\tag{19}\] In this section we prove weighted resolvent bounds for the self-adjoint realization of the above operator (which again will be denoted by \(P\)) on the Hilbert space \(L^2({\mathbb{R}}^d)\). We have

Proof. We prove ?? using the Carleman estimate ?? . We keep the same notations as in the previous section. Clearly, it suffices to prove ?? for \(0<s-\frac{1}{2}\ll 1\), since this would imply the estimate for all \(s>\frac{1}{2}\). In Appendix 8 we show that, in the sense of distributions on \({\mathbb{R}}^d\), the operator \(P\) acts on \(u\) in the domain \(D(P) \subseteq H^1({\mathbb{R}}^d)\) by \[Pu =-\Delta u+i\nabla\cdot (bu)+ib\cdot\nabla u+\widetilde{V}u,\] where \(\widetilde{V}=V+|b|^2\). Here, \(\nabla \cdot (bu)\) is defined distributionally by \((\nabla \cdot (bu), v) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =-(u, b \cdot \nabla v)\), where \((\cdot, \cdot)\) denotes distributional pairing. We note that \(u \mapsto \nabla \cdot (bu)\) a bounded mapping from \(L^2({\mathbb{R}}^d)\) to \(H_h^{-1}({\mathbb{R}}^d)\). Given \(g \in C^\infty_0(\mathbb{R}^d)\), set \[f=(P-\lambda^2\pm i\varepsilon)^{-1}g \in D(P) \cap H^1({\mathbb{R}}^d),\qquad f_1=e^{\tau\varphi}f \in H^1({\mathbb{R}}^d).\] Both \(P\) and \(P_{0, \varphi} = -e^{\tau \varphi} \Delta e^{-\tau \varphi}:H_h^1({\mathbb{R}}^d) \to H_h^{-1}({\mathbb{R}}^d)\) are bounded. As members of \(H^{-1}_h({\mathbb{R}}^d)\), \[\begin{align} P_{0, \varphi} f_1 - e^{\tau \varphi} Pf &= -e^{\tau \varphi} (i \nabla \cdot (bf) + ib \cdot \nabla f + \tilde{V} f) \\ &= i\nabla\cdot (bf_1) -ib\cdot\nabla f_1 -\widetilde{V} f_1 +2i\tau\nabla\varphi\cdot b f_1. \end{align}\] By the definition of \(\varphi\), we have \(\nabla \varphi = O(\langle r \rangle^{-2s})\), and if we take \(s > \tfrac{1}{2}\) small enough so that \(2s < \rho\) with \(\rho\) as in [3461], then \[\label{eq:3463} \left\|\langle x\rangle^{s}\left(P_{0, \varphi} f_1 - e^{\tau \varphi} Pf\right) \right\|_{H_h^{-1}}\lesssim h^{-1}\|\langle x\rangle^{-s}f_1\|_{H_h^1}.\tag{20}\] We are going to use the estimate ?? with \(f\) replaced by \(f_1\). Note that \(f\) satisfies the required hypothesis of Proposition 5 because \[\begin{align} \langle x \rangle^{s}(P_{0, \varphi} &- \lambda^2 \pm i \varepsilon) e^{\tau \varphi} (P - \lambda^2 \pm i\varepsilon)^{-1} g\\ &= \langle x \rangle^{s} e^{\tau \varphi }( -\Delta - \lambda^2 \pm i \varepsilon) (P - \lambda^2 \pm i\varepsilon)^{-1} g\\ &= \langle x \rangle^{s} e^{\tau \varphi} g + \langle x \rangle^{s} e^{\tau \varphi} ( i \nabla \cdot b + ib \cdot \nabla + \widetilde{V} ) (P- \lambda^2 \pm i\varepsilon)^{-1}g \in H^{-1}({\mathbb{R}}^d). \end{align}\] By (?? ) and (20 ) we get \[\begin{align} \|\langle x\rangle^{-s}f_1\|_{H_h^1}&\lesssim h\tau^{-1/2}\|\langle x\rangle^{s} e^{\tau \varphi}(P-\lambda^2\pm i\varepsilon)f\|_{H_h^{-1}}\\ &+\tau^{-1/2}\|\langle x\rangle^{-s}f_1\|_{L^2} +A^{\ell}(\varepsilon h)^{1/2}\|f_1\|_{L^2}. \end{align}\] We can absorb the second term in the right-hand side of the above inequality by taking \(\tau\) large enough independent of \(\lambda\). Since \(h<\lambda^{-1}\), this leads to \[\|\langle x\rangle^{-s}f_1\|_{H_h^1}\lesssim \lambda^{-1}\|\langle x\rangle^{s} e^{\tau \varphi}(P-\lambda^2\pm i\varepsilon)f\|_{H_h^{-1}} +\varepsilon^{1/2}\lambda^{-1/2}\|f_1\|_{L^2},\] which in turn implies \[\label{eq:3464} \|\langle x\rangle^{-s}f\|_{H_h^1}\lesssim \lambda^{-1}\|\langle x\rangle^{s} (P-\lambda^2\pm i\varepsilon)f\|_{H_h^{-1}} +\varepsilon^{1/2}\lambda^{-1/2}\|f\|_{L^2},\tag{21}\] where the implicit constant depends on \(\tau\), which is now fixed, but is indepedent of \(\lambda\). On the other hand, the symmetry of the operator \(P\) on the Hilbert space \(L^2(\mathbb{R}^d)\) gives \[\label{eq:3465} \begin{align} \varepsilon\|f\|_{L^2}^2&=\left| {\rm Im}\,\left\langle(P-\lambda^2\pm i\varepsilon)f,f\right\rangle_{L^2} \right| \\ &\le\left|\left\langle \langle x\rangle^{s}g,\langle x\rangle^{-s}f\right\rangle_{L^2}\right|\\ &\le \gamma\lambda\|\langle x\rangle^{-s}f\|_{H_h^1}^2+ \gamma^{-1}\lambda^{-1}\|\langle x\rangle^{s}g\|_{H_h^{-1}}^2 \end{align}\tag{22}\] for every \(\gamma>0\). Combining 21 , 22 and taking \(\gamma\) small enough independent of \(\lambda\) we obtain \[\label{eq:3466} \|\langle x\rangle^{-s}f\|_{H_h^1}\lesssim\lambda^{-1} \|\langle x\rangle^{s}g\|_{H_h^{-1}}.\tag{23}\] It is easy to see that 23 is equivalent to ?? .  \(\Box\)

Denote \(\mathbb{C}^-:=\{\lambda\in \mathbb{C}:{\rm Im}\,\lambda<0\}\) and \(\mathcal{L}=\mathbb{C}\) if \(d\) is odd, while \[\mathcal{L}=\left\{\lambda\in\mathbb{C}:-\frac{3\pi}{2}<\arg(\lambda)<\frac{\pi}{2}\right\}\] if \(d\) is even. Also, given a parameter \(\gamma>0\), set \(\mathcal{L}_\gamma=\{\lambda\in\mathcal{L}:{\rm Im}\,\lambda<\gamma\}\). In Proposition 7 below, we combine ?? with estimates for the free resolvent (reviewed in Appendix 10) to construct an analytic continuation of the operator valued function \(\mu (P - \lambda^2)^{-1}\mu : L^2({\mathbb{R}}^d) \to L^2({\mathbb{R}}^d)\) from \({\mathbb{C}}^{-}\) into \(\mathcal{L}_{\gamma}\), for \(\gamma\) small enough.

From ?? and Lemma 7, we obtain the following bounds on the \(\lambda\)-derivatives of \(\mu(P - \lambda^2)^{-1} \mu\), which are key to our proof of wave decay in Section 6.

Proof of Proposition 7. Denote by \(P_0\) the self-adjoint realization of \(-\Delta\) on \(L^2(\mathbb{R}^d)\). Let \(\lambda\in\mathbb{C}^-\) and denote by \(I\) the identity operator. We begin from two resolvent identities, \[\begin{align} & (P- \lambda^2)^{-1} (\widetilde{V} + i\nabla\cdot b+ib\cdot\nabla) = I - (P- \lambda^2)^{-1} (P_0 - \lambda^2) \quad \text{on}\quad H^2({\mathbb{R}}^d), \\ & (P_0- \lambda^2)^{-1} (\widetilde{V} + i\nabla\cdot b+ib\cdot\nabla) = -I + (P_0- \lambda^2)^{-1} (P - \lambda^2) \quad \text{on}\quad D(P), \end{align}\] the first of which we prove in detail in Appendix 8. These yield \[\label{eq:3469} \begin{align} &(P-\lambda^2)^{-1}-(P_0-\lambda^2)^{-1}\\ &=-(P_0-\lambda^2)^{-1}(\widetilde{V}+i\nabla\cdot b+ib\cdot\nabla)(P-\lambda^2)^{-1}\\ &=-(P-\lambda^2)^{-1}(\widetilde{V}+i\nabla\cdot b+ib\cdot\nabla)(P_0-\lambda^2)^{-1}. \end{align}\tag{24}\] Let \(z\in\mathbb{C}^-\). By 24 , we get \[\label{eq:34610} \begin{align} &(P-\lambda^2)^{-1}-(P-z^2)^{-1}\\ &=(\lambda^2-z^2)(P-z^2)^{-1}(P-\lambda^2)^{-1}\\ &=L^\sharp(z)((P_0-\lambda^2)^{-1}-(P_0-z^2)^{-1})L^\flat(\lambda), \end{align}\tag{25}\] where \[\begin{align} & L^\sharp=I-(P-z^2)^{-1}(\widetilde{V}+i\nabla\cdot b+ib\cdot\nabla),\\ & L^\flat=I-(\widetilde{V}+i\nabla\cdot b+ib\cdot\nabla)(P-\lambda^2)^{-1}. \end{align}\] Multiplying both sides of (25 ) by \(\mu\) we get \[\label{eq:34611} \begin{align} &\mu(P-\lambda^2)^{-1}\mu-\mu(P-z^2)^{-1}\mu\\ &=\sum_{\ell_1=0}^1\sum_{\ell_2=0}^1L^\sharp_{\ell_1}(z)\mu^{1-\ell_1}(-i\mu^{-1}b\cdot\nabla)^{\ell_1}((P_0-\lambda^2)^{-1}-(P_0-z^2)^{-1})(-i\nabla\cdot b\mu^{-1})^{\ell_2}\mu^{1-\ell_2}L^\flat_{\ell_2}(\lambda), \end{align}\tag{26}\] where \[\begin{align} & L_0^\sharp=I-\mu(P-z^2)^{-1}(\widetilde{V}+i\nabla\cdot b)\mu^{-1},\\ & L_1^\sharp=\mu(P-z^2)^{-1}\mu,\\ & L_0^\flat=I-\mu^{-1}(\widetilde{V}+ib\cdot\nabla)(P-\lambda^2)^{-1}\mu,\\ & L_1^\flat=\mu(P-\lambda^2)^{-1}\mu, \end{align}\] are bounded operators on \(L^2(\mathbb{R}^d)\). We now let the operator \(\mu^{-1}ib\cdot\nabla\) act on the left side of (25 ) and multiply the right side by \(\mu\). We get \[\label{eq:34612} \mu^{-1}ib\cdot\nabla(P-\lambda^2)^{-1}\mu=T_1(\lambda, z)+T_2(\lambda,z)\mu(P-\lambda^2)^{-1}\mu+T_3(\lambda,z) \mu^{-1}ib\cdot\nabla(P-\lambda^2)^{-1}\mu,\tag{27}\] where \[\begin{align} & T_1=\mu^{-1}ib\cdot\nabla(P-z^2)^{-1}\mu -\sum_{\ell_1=0}^1 \widetilde{L}^\sharp_{\ell_1}(z)\mu^{1-\ell_1}(i\mu^{-1}b\cdot\nabla)^{\ell_1}((P_0 - \lambda^2)^{-1} - (P_0 - z^2)^{-1})\mu ,\\ & T_2=\sum_{\ell_1=0}^1\sum_{\ell_2=0}^1\widetilde{L}^\sharp_{\ell_1}(z)\mu^{1-\ell_1}(i\mu^{-1}b\cdot\nabla)^{\ell_1}((P_0-\lambda^2)^{-1}-(P_0-z^2)^{-1}) (\widetilde{V}\mu^{-1})^{1-\ell_2}(i\nabla\cdot b\mu^{-1})^{\ell_2},\\ & T_3=\sum_{\ell_1=0}^1\widetilde{L}^\sharp_{\ell_1}(z)\mu^{1-\ell_1} (i\mu^{-1}b\cdot\nabla)^{\ell_1}((P_0-\lambda^2)^{-1}-(P_0-z^2)^{-1})\mu,\\ &\widetilde{L}_0^\sharp=\mu^{-1}ib\cdot\nabla(P-z^2)^{-1}(\widetilde{V}+i\nabla\cdot b)\mu^{-1},\\ &\widetilde{L}_1^\sharp=-I+\mu^{-1}ib\cdot\nabla(P-z^2)^{-1}\mu. \end{align}\] Fix \(z \in {\mathbb{C}}^-\) and consider the above operators as functions of \(\lambda\). Due to the exponential decay 3 , the operators \(\widetilde{L}_0^\sharp\) and \(\widetilde{L}_1^\sharp\) are bounded on \(L^2({\mathbb{R}}^d)\). Furthermore, the operators \((i\mu^{-1}b\cdot\nabla)^{\ell_1}(P_0-\lambda^2)^{-1}\mu\), \(\ell_1=0,1\) are compact and, in view of Lemma 9, extend holomorphically to \(\mathcal{L}_{\gamma_0}\) for some constant \(\gamma_0>0\). Hence \(T_3(\lambda,z)\) is a family of compact operators, analytic in \(\mathcal{L}_{\gamma_0}\). Therefore, since \(T_3(z,z)\equiv 0\), by the Fredholm theorem we conclude \((I-T_3(\lambda,z))^{-1}\) exists as a meromorphic in \(\mathcal{L}_{\gamma_0}\) operator-valued function. Thus by (27 ), still for \(\lambda\in\mathbb{C}^-\), we get \[\label{eq:34613} \mu^{-1}ib\cdot\nabla(P-\lambda^2)^{-1}\mu=(I-T_3)^{-1}T_1+(I-T_3)^{-1}T_2\mu(P-\lambda^2)^{-1}\mu.\tag{28}\] By (26 ) and (28 ), \[\label{eq:34614} \mu(P-\lambda^2)^{-1}\mu=F_1(\lambda,z)+F_2(\lambda,z)\mu(P-\lambda^2)^{-1}\mu,\tag{29}\] where \[\begin{align} F_1&=\mu(P-z^2)^{-1}\mu + \sum_{\ell_1 = 0}^1 L^\sharp_{\ell_1}(z)\mu^{1-\ell_1}(-i\mu^{-1}b\cdot\nabla)^{\ell_1} ((P_0-\lambda^2)^{-1}-(P_0-z^2)^{-1})\mu \\ &-\sum_{\ell_1=0}^1L^\sharp_{\ell_1}(z)\mu^{1-\ell_1}(-i\mu^{-1}b\cdot\nabla)^{\ell_1} ((P_0-\lambda^2)^{-1}-(P_0-z^2)^{-1})\mu(I-T_3)^{-1}T_1,\\ F_2&=\sum_{\ell_1=0}^1\sum_{\ell_2=0}^1L^\sharp_{\ell_1}(z)\mu^{1-\ell_1}(-i\mu^{-1}b\cdot\nabla)^{\ell_1} \\ &\cdot ((P_0-\lambda^2)^{-1}-(P_0-z^2)^{-1})(-i\nabla\cdot b\mu^{-1})^{\ell_2}(-\widetilde{V}\mu^{-1})^{1-\ell_2}\\ &-\sum_{\ell_1=0}^1L^\sharp_{\ell_1}(z)\mu^{1-\ell_1}(-i\mu^{-1}b\cdot\nabla)^{\ell_1} ((P_0-\lambda^2)^{-1}-(P_0-z^2)^{-1})\mu(I-T_3)^{-1}T_2. \end{align}\] It is easy to see that the operator \(F_2\) sends \(L^2(\mathbb{R}^d)\) into \(H^1(\mathbb{R}^d)\). Therefore \(F_2\) is a meromorphic (in \(\lambda\in\mathcal{L}_{\gamma_0}\)) family of compact operators on \(L^2({\mathbb{R}}^d)\). Since \(F_2(z,z)\equiv 0\), this implies that \((I-F_2)^{-1}\) and \(F_1\) are meromorphic operator-valued functions in \(\mathcal{L}_{\gamma_0}\) and, by (29 ), we have \[\label{eq:34615} \mu(P-\lambda^2)^{-1}\mu=(I-F_2(\lambda,z))^{-1}F_1(\lambda,z).\tag{30}\] Thus we conclude that \[\mu(P-\lambda^2)^{-1}\mu:L^2({\mathbb{R}}^d) \to L^2({\mathbb{R}}^d)\] extends meromorphically from \(\mathbb{C}^-\) to \(\mathcal{L}_{\gamma_0}\). Note also that in view of the resolvent estimate ?? , the identity (30 ) extends to all \(z\in \mathbb{R}\), \(z\neq 0\).

Let now \(0<{\rm Im}\,\lambda<\gamma_0\), \(z={\rm Re}\,\lambda\), \(|z|\ge\delta\), \(0<\delta\ll 1\) being arbitrary. It follows from the resolvent estimate (?? ) that \[\|\widetilde{L}_\ell^\sharp(z)\|\lesssim |z|^{1-\ell},\quad \ell=0,1,\] which together with (?? ) imply \[\label{eq:34616} \|T_3(\lambda,z)\|\lesssim {\rm Im}\,\lambda\le 1/2,\tag{31}\] if \({\rm Im}\,\lambda\le\gamma_1\) with some constant \(0<\gamma_1<\gamma_0\). By (?? ) and (?? ) we also have \[\label{eq:34617} \|T_j(\lambda,z)\|\lesssim |z|^{j-1},\quad j=1,2,\tag{32}\] \[\label{eq:34618} \|L_\ell^\sharp(z)\|\lesssim |z|^{-\ell},\quad \ell=0,1.\tag{33}\] By (31 ), (32 ) and (33 ) together with (?? ), \[\label{eq:34619} \|F_1(\lambda,z)\|\lesssim |z|^{-1},\tag{34}\] \[\label{eq:34620} \|F_2(\lambda,z)\|\lesssim {\rm Im}\,\lambda\le 1/2,\tag{35}\] if \({\rm Im}\,\lambda\le\gamma_2\) with some constant \(0<\gamma_2<\gamma_1\). By (30 ) and (35 ) we conclude \(\mu(P-\lambda^2)^{-1}\mu\) is analytic in \(\{\lambda\in\mathcal{L}_{\gamma_2},\,|{\rm Re}\,\lambda|\ge\delta\}\). In odd dimensions, if 6 holds, then \(\mu(P-\lambda^2)^{-1}\mu\) is analytic in \(\mathcal{L}_{\gamma_2}\) since 6 implies \(\lambda = 0\) is not a pole. The estimate (?? ) with \(\ell=0\), \(\gamma=\gamma_2\), follows from (30 ), (34 ) and (35 ). The estimate (?? ) with \(\ell=1\) is obtained by combining ?? with \(\ell=0\), the first identity in 24 , 28 , and ?? .  \(\Box\)

4 Resolvent bounds in the exterior of a non-trapping obstacle↩︎

Let \(\mathcal{O}\subset\mathbb{R}^d\), \(d\ge 2\), be a bounded domain with smooth boundary such that \(\Omega=\mathbb{R}^d\setminus\mathcal{O}\) is connected. Denote by \(P\) the Dirichlet self-adjoint realization of \(-\Delta+V\) on the Hilbert space \(L^2(\Omega)\), where \(V\in L^\infty(\Omega)\) is a real-valued potential satisfying \[\label{eq:4461} |V(x)|\le C\langle x\rangle^{-\rho}, \quad C>0,\rho>1.\tag{36}\] We have

Proof. In view of the coercivity of the operator \(\widetilde{P}\), the bound (4 ) implies \[\label{eq:4463} \left\|\chi\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\chi\right\| \lesssim\lambda^{|\alpha|+|\beta|-1},\quad\lambda\ge\lambda_0,\tag{37}\] for all multi-indices \(\alpha\) and \(\beta\) such that \(|\alpha|\le 1\) and \(|\beta|\le 1\). Let us see that (37 ) implies the weighted resolvent bounds \[\label{eq:4464} \left\|\langle x\rangle^{-s}\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\langle x\rangle^{-s}\right\| \lesssim\lambda^{|\alpha|+|\beta|-1},\quad\lambda\ge\lambda_0,\tag{38}\] for every \(s>1/2\) and all multi-indices \(\alpha\) and \(\beta\) such that \(|\alpha|\le 1\) and \(|\beta|\le 1\). To this end we, use the fact that (38 ) holds for the operator \(P_0\), the self-adjoint realization of \(-\Delta\) on \(L^2(\mathbb{R}^d)\) (see Lemma 8). Let \(\eta, \, \chi \in C^\infty(\mathbb{R}^d)\) be of compact support such that \(\eta =1\) on \(\mathcal{O}\) and \(\chi=1\) on supp\(\,\eta\). We have the identity \[(P_0-\lambda^2\pm i\varepsilon)(1-\eta)(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}= [\Delta,\eta](\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}+1-\eta,\] which implies \[\label{eq:4465} (1-\eta)(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}=(P_0-\lambda^2\pm i\varepsilon)^{-1}[\Delta,\eta](\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}+(P_0-\lambda^2\pm i\varepsilon)^{-1}(1-\eta).\tag{39}\] Similarly, \[\label{eq:4466} (\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}(1-\eta)=(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}[\Delta,\eta](P_0-\lambda^2\pm i\varepsilon)^{-1}+(1-\eta)(P_0-\lambda^2\pm i\varepsilon)^{-1}.\tag{40}\] By 37 , (39 ), and (40 ), \[\begin{align} &\left\|\langle x\rangle^{-s}\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\langle x\rangle^{-s}\right\|\le \left\|\chi\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\langle x\rangle^{-s}\right\|\\ &+\left\|\langle x\rangle^{-s}(1-\eta)\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\langle x\rangle^{-s}\right\|\\ &\lesssim \left\|\chi\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\langle x\rangle^{-s}\right\|+\lambda^{|\alpha|}\left\|\chi(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\langle x\rangle^{-s}\right\|+\lambda^{|\alpha|+|\beta|-1}\\ &\lesssim \left\|\chi\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\chi\right\|+\lambda^{|\alpha|}\left\|\chi(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\chi\right\|+\lambda^{|\alpha|+|\beta|-1}\\ &+\left\|\chi\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta(1-\eta)\langle x\rangle^{-s}\right\|+\lambda^{|\alpha|}\left\|\chi(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta(1-\eta)\langle x\rangle^{-s}\right\|+\lambda^{|\alpha|+|\beta|-1}\\ &\lesssim \left\|\chi\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\beta\chi\right\|+\lambda^{|\beta|}\left\|\chi\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \chi\right\|\\ &+\lambda^{|\alpha|+|\beta|}\left\|\chi(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} \chi\right\|+\lambda^{|\alpha|+|\beta|-1}\lesssim \lambda^{|\alpha|+|\beta|-1}. \end{align}\]

We now derive (?? ) from (38 ) for large \(\lambda\). To this end we use the resolvent identity \[\label{eq:4467} (I+K(\lambda))\langle x\rangle^{-s}(P-\lambda^2\pm i\varepsilon)^{-1}\langle x\rangle^{-s} =\langle x\rangle^{-s}(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}\langle x\rangle^{-s},\tag{41}\] where \[K(\lambda)=\langle x\rangle^{-s}(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}\langle x\rangle^{s}V.\] If \(1/2<s\le\rho/2\), by (38 ) we get \[\label{eq:4468} \|K(\lambda)\|\le C\lambda^{-1}\le 1/2,\tag{42}\] for \(\lambda\gg 1\). It follows from (38 ), (41 ) and (42 ) that there is a constant \(\lambda_1>\lambda_0\) such that (?? ) with \(\alpha=\beta=0\) holds for \(\lambda\ge\lambda_1\). In the general case (?? ) follows from the identities \[\label{eq:4469} \begin{align} &\langle x\rangle^{-s}\partial_x^\alpha(P-\lambda^2\pm i\varepsilon)^{-1}\partial_x^\beta\langle x\rangle^{-s}- \langle x\rangle^{-s}\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}\partial_x^\beta\langle x\rangle^{-s}\\ &=-\langle x\rangle^{-s}\partial_x^\alpha(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1} V(P-\lambda^2\pm i\varepsilon)^{-1}\partial_x^\beta\langle x\rangle^{-s}\\ &=-\langle x\rangle^{-s}\partial_x^\alpha(P-\lambda^2\pm i\varepsilon)^{-1} V(\widetilde{P}-\lambda^2\pm i\varepsilon)^{-1}\partial_x^\beta\langle x\rangle^{-s}, \end{align}\tag{43}\] together with (38 ) and (?? ) with \(\alpha=\beta=0\).

Next we prove (?? ) as well as (38 ) for \(\delta\le\lambda\le\lambda_1\), \(0<\delta\ll 1\) being arbitrary, by using the Carleman estimate (?? ). We keep the same notations as in Section 2. Given a function \(g \in L^2(\Omega)\) such that \(\langle x\rangle^{s}g\in L^2(\Omega)\), set \[f=(P-\lambda^2\pm i\varepsilon)^{-1}g.\] Clearly, \(f|_{\partial\Omega}=0\). Fix \(a\gg 1\) be such that \(\mathcal{O}\subset B_a:=\{x\in \mathbb{R}^d:|x|\le a\}\). Choose functions \(\psi_a,\widetilde{\psi}_a\in C_0^\infty(\mathbb{R}^d)\) such that \(\widetilde{\psi}_a(x)=1\) for \(|x|\le a+1\), \(\widetilde{\psi}_a(x)=0\) for \(|x|\ge a+2\), \(\psi_a(x)=1\) for \(|x|\le a+3\), \(\psi_a(x)=0\) for \(|x|\ge a+4\). Now Theorem 2.1 of [26] applied to the function \(\psi_a f\) (with \(h=1\)) leads to the estimate \[\label{eq:44610} \begin{align} \|\psi_a f\|_{H^1(\Omega)}&\lesssim \|(P-\lambda^2\pm i\varepsilon)(\psi_a f)\|_{L^2(\Omega)}\\ &\lesssim \|\psi_a g\|_{L^2(\Omega)}+\|[\Delta,\psi_a]f\|_{L^2(\Omega)}\lesssim \|\psi_a g\|_{L^2(\Omega)} +\|f\|_{H^1(B_{a+4}\setminus B_{a+3})}. \end{align}\tag{44}\] Let \(1/2<s<\min\{1,\rho/2\}\). We now use the estimate (?? ) with \(f\) replaced by \(e^{\tau\varphi}(1-\widetilde{\psi}_a)f\). Since \(h^{-1}\le \tau+\lambda_1\), we obtain the estimate \[\label{eq:44611} \begin{align} &\|\langle x\rangle^{-s}e^{\tau\varphi}(1-\widetilde{\psi}_a)f\|_{H^1(\mathbb{R}^d)}\le Ch^{-1}\|\langle x\rangle^{-s}e^{\tau\varphi}(1-\widetilde{\psi}_a)f\|_{H^1_h(\mathbb{R}^d)}\\ &\le C\tau^{-1/2}\|\langle x\rangle^{s}e^{\tau\varphi} (-\Delta-\lambda^2\pm i\varepsilon)(1-\widetilde{\psi}_a)f\|_{L^2(\mathbb{R}^d)} +C_\tau\varepsilon^{1/2}\|e^{\tau\varphi}(1-\widetilde{\psi}_a)f\|_{L^2(\mathbb{R}^d)}\\ &\le C\tau^{-1/2}\|\langle x\rangle^{s}e^{\tau\varphi} (-\Delta+V-\lambda^2\pm i\varepsilon)(1-\widetilde{\psi}_a)f\|_{L^2(\mathbb{R}^d)}\\ &+C\tau^{-1/2}\|\langle x\rangle^{-s}e^{\tau\varphi}(1-\widetilde{\psi}_a)f\|_{L^2(\mathbb{R}^d)}+ C_\tau\varepsilon^{1/2}\|e^{\tau\varphi}(1-\widetilde{\psi}_a)f\|_{L^2(\mathbb{R}^d)}. \end{align}\tag{45}\] Hereafter \(C>0\) denotes a constant, independent of \(\tau\), which may change from line to line, while \(C_\tau>0\) denotes a constant, depending on \(\tau\), which may change from line to line and whose precise value is not important in the analysis that follows. Taking \(\tau\) large enough we can absorb the second term in the right-hand side of (45 ) to obtain \[\label{eq:44612} \begin{align} &\|\langle x\rangle^{-s}e^{\tau\varphi}(1-\widetilde{\psi}_a)f\|_{H^1(\mathbb{R}^d)}\le C\|\langle x\rangle^{s}e^{\tau\varphi}(1-\widetilde{\psi}_a)g\|_{L^2(\mathbb{R}^d)}\\ &+C\|\langle x\rangle^{s}e^{\tau\varphi}[\Delta,\widetilde{\psi}_a]f\|_{L^2(\mathbb{R}^d)} +C_\tau\varepsilon^{1/2}\|e^{\tau\varphi}(1-\widetilde{\psi}_a)f\|_{L^2(\mathbb{R}^d)}\\ &\le C\|\langle x\rangle^{s}g\|_{L^2(\Omega)}+Ce^{\tau\varphi(a+2)}\|f\|_{H^1(B_{a+2}\setminus B_{a+1})} +C_\tau \varepsilon^{1/2}\|f\|_{L^2(\Omega)}. \end{align}\tag{46}\] In particular, (46 ) implies \[\label{eq:44613} \begin{align} &e^{\tau\varphi(a+3)}\|f\|_{H^1(B_{a+4}\setminus B_{a+3})}\\ &\le C\|\langle x\rangle^{s}g\|_{L^2(\Omega)}+Ce^{\tau\varphi(a+2)}\|f\|_{H^1(B_{a+2}\setminus B_{a+1})} +C_\tau\varepsilon^{1/2}\|f\|_{L^2(\Omega)}\\ &\le C\|\langle x\rangle^{s}g\|_{L^2(\Omega)}+Ce^{\tau\varphi(a+2)}\|f\|_{H^1(B_{a+4}\setminus B_{a+3})} +C_\tau\varepsilon^{1/2}\|f\|_{L^2(\Omega)}, \end{align}\tag{47}\] where we have also used (44 ). Since \(\varphi(a+3)-\varphi(a+2)>0\) is independent of \(\tau\), we can absorb the second term in the right-hand side of (47 ) by taking \(\tau\) large enough. We now fix \(\tau\). Thus we obtain \[\label{eq:44614} \|f\|_{H^1(B_{a+4}\setminus B_{a+3})}\lesssim \|\langle x\rangle^{s}g\|_{L^2(\Omega)} +\varepsilon^{1/2}\|f\|_{L^2(\Omega)}.\tag{48}\] Combining (44 ), (46 ) and (48 ) leads to \[\label{eq:44615} \|\langle x\rangle^{-s}f\|_{H^1(\Omega)}\lesssim \|\langle x\rangle^{s}g\|_{L^2(\Omega)} +\varepsilon^{1/2}\|f\|_{L^2(\Omega)}.\tag{49}\] On the other hand, the symmetry of the operator \(P\) on the Hilbert space \(L^2(\Omega)\) gives \[\label{eq:44616} \begin{align} \varepsilon\|f\|_{L^2(\Omega)}^2&= \left | {\rm Im}\:\langle (P- \lambda^2 \pm i \varepsilon)f, f \rangle_{L^2(\Omega)} \right |\\ &\le \left|\left\langle \langle x\rangle^{s}g,\langle x\rangle^{-s}f\right\rangle_{L^2(\Omega)}\right|\\ &\le \gamma\|\langle x\rangle^{-s}f\|_{L^2(\Omega)}^2+ \gamma^{-1}\|\langle x\rangle^{s}g\|_{L^2(\Omega)}^2 \end{align}\tag{50}\] for every \(\gamma>0\). Combining (49 ), (50 ) and taking \(\gamma\) small enough we obtain the estimate \[\label{eq:44617} \|\langle x\rangle^{-s}f\|_{H^1(\Omega)}\lesssim \|\langle x\rangle^{s}g\|_{L^2(\Omega)},\tag{51}\] which implies (?? ) as well as (38 ) for \(\delta\le\lambda\le\lambda_1\) and \(|\alpha|\le 1\), \(\beta=0\). For \(|\alpha|\le 1\), \(|\beta|\le 1\) the estimate (38 ) follows from the coercivity of the operator \(\widetilde{P}\), while (?? ) follows from (38 ) and the identities (43 ).  \(\Box\)

Like in the previous section, we develop the meromorphic continuation of the operator \(\mu (P - \lambda^2)^{-1} \mu : L^2(\Omega) \to L^2(\Omega)\), and establish resolvent bounds crucial for obtaining wave decay in Section 6.

Proof. We follow the same strategy as in the proof of Proposition 7. Let \(\eta\in C^\infty(\mathbb{R}^d)\) be of compact support such that \(\eta =1\) on \(\mathcal{O}\). For \(\lambda\in\mathbb{C}^-\) we have \[(P_0-\lambda^2)(1-\eta)(P-\lambda^2)^{-1}=([\Delta,\eta]-(1-\eta)V)(P-\lambda^2)^{-1}+1-\eta, \qquad \text{on } L^2(\Omega),\] which implies \[\label{eq:44619} (1-\eta)(P-\lambda^2)^{-1} =(P_0-\lambda^2)^{-1}([\Delta,\eta]-(1-\eta)V)(P-\lambda^2)^{-1}+(P_0-\lambda^2)^{-1}(1-\eta).\tag{52}\] Let \(z\in\mathbb{C}^-\). Similarly, \[\label{eq:44620} \begin{align} (P&-z^2)^{-1}(1-\eta)\\ &=(P-z^2)^{-1}([\Delta,\eta]-(1-\eta)V)(P_0-z^2)^{-1}+(1-\eta)(P_0-z^2)^{-1}, \qquad \text{on } L^2({\mathbb{R}}^d). \end{align}\tag{53}\] In view of (52 ) and (53 ), \[\begin{align} (P-\lambda^2)^{-1}-(P-z^2)^{-1}&=(\lambda^2-z^2)(P-z^2)^{-1}(P-\lambda^2)^{-1}\\ &=(\lambda^2-z^2)(P-z^2)^{-1}\eta(2-\eta)(P-\lambda^2)^{-1}\\ &+(\lambda^2-z^2)(P-z^2)^{-1}(1-\eta)^2(P-\lambda^2)^{-1}\\ &=(\lambda^2-z^2)(P-z^2)^{-1}\eta(2-\eta)(P-\lambda^2)^{-1}\\ &+(1-\eta+(P-z^2)^{-1}([\Delta,\eta]-(1-\eta)V))((P_0-\lambda^2)^{-1}\\ &-(P_0-z^2)^{-1})(1-\eta+([\Delta,\eta]-(1-\eta)V)(P-\lambda^2)^{-1}). \end{align}\] Multiplying both sides of this identity by \(\mu\) we get \[\label{eq:44621} \begin{align} \mu(P-\lambda^2)^{-1}\mu-\mu(P-z^2)^{-1}\mu&=(\lambda^2-z^2)\mu(P-z^2)^{-1}\eta(2-\eta)(P-\lambda^2)^{-1}\mu\\ &+Q_1(z)(\mu(P_0-\lambda^2)^{-1}\mu-\mu(P_0-z^2)^{-1}\mu)Q_2(\lambda), \end{align}\tag{54}\] where \[\begin{align} & Q_1(z)=1-\eta+\mu(P-z^2)^{-1}([\Delta,\eta]-(1-\eta)V)\mu^{-1},\\ & Q_2(\lambda)=1-\eta+\mu^{-1}([\Delta,\eta]-(1-\eta)V)(P-\lambda^2)^{-1}\mu. \end{align}\] We rewrite (54 ) in the form \[\label{eq:44622} \begin{align} &(I-K(\lambda,z))\mu(P-\lambda^2)^{-1}\mu=\mu(P-z^2)^{-1}\mu\\ &+Q_1(z)(\mu(P_0-\lambda^2)^{-1}\mu-\mu(P_0-z^2)^{-1}\mu)(1-\eta), \end{align}\tag{55}\] where the operator \[\begin{align} K(\lambda,z)&=(\lambda^2-z^2)\mu(P-z^2)^{-1}\eta(2-\eta)\mu^{-1}\\ &+Q_1(z)(\mu(P_0-\lambda^2)^{-1}-\mu(P_0-z^2)^{-1})([\Delta,\eta]-(1-\eta)V)\mu^{-1} \end{align}\] sends \(L^2(\Omega)\) into \(H^1(\Omega)\) and extends analytically in \(\lambda\in \mathcal{L}_{\gamma_0}\) in view of Lemma 9. Therefore \(K(\lambda,z)\) is a family of compact operators on \(L^2(\Omega)\), analytic in \(\mathcal{L}_{\gamma_0}\). Since \(K(z,z)\equiv 0\), by the Analytic Fredholm theorem \((I-K(\lambda,z))^{-1}\) exists as a meromorphic in \(\mathcal{L}_{\gamma_0}\) operator-valued function. By (55 ) we get that \(\mu(P-\lambda^2)^{-1}\mu\) extends meromorphically from \(\mathbb{C}^-\) to \(\mathcal{L}_{\gamma_0}\). Moreover, the identity (55 ) extends to all \(\lambda\in \mathcal{L}_{\gamma_0}\) as well as to all \(z\in\mathbb{R}\), \(z\neq 0\). Let now \(0<{\rm Im}\,\lambda<\gamma_0\), \(z={\rm Re}\,\lambda\), \(|z|\ge\delta\), \(0<\delta\ll 1\) being arbitrary. It follows from (?? ) that \[\label{eq:44623} \|Q_1(z)\|\lesssim 1.\tag{56}\] By (?? ), (56 ) and (?? ), \[\label{eq:44624} \|K(\lambda,z)\|\lesssim {\rm Im}\,\lambda\le 1/2\tag{57}\] for \({\rm Im}\,\lambda\le\gamma_1\) with some constant \(0<\gamma_1<\gamma_0\). Thus, by (55 ) and (57 ) we obtain that \(\mu(P-\lambda^2)^{-1}\mu\) extends analytically to \(\{\lambda\in\mathcal{L}_{\gamma_1},\,|{\rm Re}\,\lambda|\ge\delta\}\). In odd dimensions \(\mu(P-\lambda^2)^{-1}\mu\) is analytic in \(\mathcal{L}_{\gamma_1}\) since the condition (6 ) implies that \(\lambda = 0\) is not a pole. Also from 55 it is easy to see that the analog of (?? ) is valid in this case, whence (?? ) follows from this fact and Lemma 7.  \(\Box\)

5 Low-frequency resolvent bounds↩︎

Let \(P: L^2({\mathbb{R}}^d) \to L^2({\mathbb{R}}^d)\) be the self-adjoint operator from Section 3. In this section we will suppose that \[\label{eq:5461} 0\le V(x)\le C\langle x\rangle^{-\rho}, \quad |b(x)|\le C\langle x\rangle^{-\rho},\tag{58}\] with constants \(C>0\), \(\rho>\max\{3,\frac{d}{2}\}\). We have the following

Proof. Given a function \(g \in L^2({\mathbb{R}}^d)\) such that \(\langle x\rangle^{s}g\in L^2({\mathbb{R}}^d)\), set \[f=(P-\lambda^2\pm i\varepsilon)^{-1}g.\] Let \(a\gg 1\) be a parameter independent of \(\lambda\) and choose \(\chi_a\in C_0^\infty(\mathbb{R}^d; [0,1])\) such that \(\chi_a(x)=1\) for \(|x|\le 3a\), \(\chi_a(x)=0\) for \(|x|\ge 4a\), and \(\partial_x^\alpha\chi_a(x)=O(a^{-|\alpha|})\).

For the rest of the proof \(\|\cdot\|\) and \(\langle\cdot,\cdot\rangle\) denote the norm and the scalar product in \(L^2(\mathbb{R}^d)\). We have \[(P-\lambda^2\pm i\varepsilon)(\chi_af)=\chi_ag+[P,\chi_a]f.\] Hence \[\begin{align} {\rm Re}\,\left\langle \chi_ag+[P,\chi_a]f,\chi_af\right\rangle &={\rm Re}\,\left\langle P\chi_af,\chi_af\right\rangle -\lambda^2\|\chi_af\|^2\\ &=\|(i\nabla+b)\chi_af\|^2+\left\langle V\chi_af,\chi_af\right\rangle-\lambda^2\|\chi_af\|^2\\ &\ge\|(i\nabla+b)\chi_af\|^2-\lambda^2\|\chi_af\|^2. \end{align}\] Thus we obtain \[\label{eq:5463} \|(i\nabla+b)\chi_af\|\le (\lambda+\gamma)\|\chi_af\|+\gamma^{-1}\|\chi_ag\|+\gamma^{-1}\|[P,\chi_a]f\|\tag{59}\] for every \(\gamma>0\). On the other hand, by the Poincaré inequality (?? ) we have \[\label{eq:5464} \|\chi_af\|\le Ca\|(i\nabla+b)\chi_af\|.\tag{60}\] We now combine (59 ) and (60 ). Choosing \(\gamma=a^{-1}\gamma_0\) with \(\gamma_0>0\) small enough independent of \(a\), and \(\lambda >0\) small enough depending on \(a\), we arrive at \[\label{eq:5465} a^{-1}\|\chi_af\|+\|(i\nabla+b)\chi_af\|\le Ca\|\chi_ag\|+Ca\|[P,\chi_a]f\|.\tag{61}\] On the other hand, using the resolvent identity (24 ) we obtain \[\label{eq:5466} [P,\chi_a]f=[P,\chi_a](P_0-\lambda^2\pm i\varepsilon)^{-1}g-[P,\chi_a](P_0-\lambda^2\pm i\varepsilon)^{-1} (\widetilde{V}+i\nabla\cdot b+ib\cdot\nabla)f.\tag{62}\] Observe now that \([P,\chi_a]\) is supported in \(3a\le|x|\le 4a\) and \[[P,\chi_a]=[-\Delta,\chi_a]+2ib\cdot\nabla\chi_a=-\Delta\chi_a-2\nabla\chi_a\cdot\nabla+2ib\cdot\nabla\chi_a =O(a^{-1})\cdot\nabla+O(a^{-2}).\] Hence, in view of Lemma 8, given \(0<\epsilon\ll 1\) and \(h \in L^2({\mathbb{R}}^d)\) with \(\langle x \rangle^{1 + \epsilon} h \in L^2({\mathbb{R}}^d)\), we have (with \(|\alpha| \le 1\)), \[\label{eq:5467} \begin{align} &\left\|[P,\chi_a](P_0-\lambda^2\pm i\varepsilon)^{-1} \partial_x^\alpha h\right\|\\ &\lesssim \sum_{j=0}^1a^{-1+j/2+\epsilon}\left\|\langle x\rangle^{j/2-1-\epsilon}\nabla^j(P_0-\lambda^2 \pm i\varepsilon)^{-1} \partial_x^\alpha \langle x\rangle^{j/2-1-\epsilon}\right\|\|\langle x\rangle^{-j/2+1+\epsilon}h\|\\ &\lesssim \sum_{j=0}^1a^{-1+j/2+\epsilon}\|\langle x\rangle^{-j/2+1+\epsilon}h\|. \end{align}\tag{63}\] Choose a function \(\widetilde{\chi}_a\in C_0^\infty(\mathbb{R}^d)\) such that \(\widetilde{\chi}_a(x)=1\) for \(|x|\le a\), \(\widetilde{\chi}_a(x)=0\) for \(|x|\ge 2a\), and \(\partial_x^\alpha\widetilde{\chi}_a(x)=O(a^{-|\alpha|})\). We will now bound the norms of the functions \[\begin{align} & f_1:=[P,\chi_a](P_0-\lambda^2\pm i\varepsilon)^{-1} (\widetilde{V}+i\nabla\cdot b+ib\cdot\nabla)(1-\widetilde{\chi}_a)f,\\ & f_2:=[P,\chi_a](P_0-\lambda^2\pm i\varepsilon)^{-1} (\widetilde{V}+i\nabla\cdot b+ib\cdot\nabla)\widetilde{\chi}_af. \end{align}\] We bound the norm of \(f_1\) by applying (63 ) with \(\alpha = 0\) to the function \(h = (\widetilde{V}+ib\cdot\nabla)(1-\widetilde{\chi}_a)f\), and applying (63 ) with \(|\alpha| = 1\) to the entries of \(b(1 -\tilde{\chi}_a)f\). We get \[\label{eq:5468} \|f_1\|\lesssim a^{1+3\epsilon-\rho}\sum_{j=0}^1\|\langle x\rangle^{-1-\epsilon}\nabla^j((1-\widetilde{\chi}_a)f)\|.\tag{64}\] To bound the norm of \(f_2\) we will use that the kernel of the free resolvent \((P_0-\lambda^2\pm i\varepsilon)^{-1}\) is of the form \(z^{d-2}E_d^\pm(z|x-y|)\), where \(z^2=\lambda^2\mp i\varepsilon\), \(\pm{\rm Im}\,z>0\), and the function \(E_d^\pm(\zeta)\) is given in terms of the Hankel functions by the formula \[\label{eq:5469} E_d^\pm(\zeta)=C_d\zeta^{-\frac{d-2}{2}}H^\pm_{\frac{d-2}{2}}(\zeta).\tag{65}\] It is well-known that \[\label{eq:54610} \left|\partial_\zeta^kE_d^\pm(\zeta)\right|\lesssim|\zeta|^{-d+2-k} \quadfor\quad |\zeta|\le 1,\quad k=0,1,2.\tag{66}\] Observe now that if \(x\in {\rm supp}\,[P,\chi_a]\), \(y\in {\rm supp}\,\widetilde{\chi}_a\), then \(a\le|x-y|\le 6a\). Hence we can arrange \(|z||x-y|\le 1\) by taking \(\lambda\) smaller if necessary, and by taking \(\varepsilon\) small, so that \(|z|a = (\lambda^4 + \varepsilon^2)^{1/4} a \ll 1\). Therefore, for such \(x\), \(y\) and \(z\) we derive from (66 ): \[\label{eq:54611} \left|z^{d-2}\partial_x^{j_1}\partial_y^{j_2}E^\pm_d(z|x-y|)\right|\lesssim a^{-d+2-j_1-j_2},\tag{67}\] where \(j_1,j_2\in\{0,1\}\). By (67 ), \[|f_2|\lesssim a^{-d}\sum_{j=0}^1\|\langle x\rangle^{-\rho}\nabla^j(\widetilde{\chi}_af)\|_{L^1}\lesssim a^{-d}\sum_{j=0}^1\|\nabla^j(\widetilde{\chi}_af)\|,\] where we have used that \(\langle x\rangle^{-\rho}\in L^2\). Hence \[\label{eq:54612} \|f_2\|^2=\int_{3a\le|x|\le 4a}|f_2|^2dx \lesssim a^{-d}\sum_{j=0}^1\|\nabla^j(\widetilde{\chi}_af)\|^2.\tag{68}\] By (62 ), (63 ) with \(\ell=0\), (64 ) and (68 ), \[\label{eq:54613} \begin{align} & \|[P,\chi_a]f\|\lesssim a^{-1/2+\epsilon}\|\langle x\rangle^{1+\epsilon}g\|\\ &+a^{1+3\epsilon-\rho}\sum_{j=0}^1\|\langle x\rangle^{-1-\epsilon}\nabla^j((1-\widetilde{\chi}_a)f)\|+ a^{-d/2}\sum_{j=0}^1\|\nabla^j(\widetilde{\chi}_af)\|. \end{align}\tag{69}\] By (61 ) and (69 ), \[\label{eq:54614} \begin{align} & a^{-1}\|\chi_af\|+\|(i\nabla+b)\chi_af\|\lesssim a\|\chi_ag\|+a^{1/2+\epsilon}\|\langle x\rangle^{1+\epsilon}g\|\\ & +a^{2+3\epsilon-\rho}\sum_{j=0}^1\|\langle x\rangle^{-1-\epsilon}\nabla^j((1-\widetilde{\chi}_a)f)\|+ a^{-d/2+1}\sum_{j=0}^1\|\nabla^j(\chi_af)\|. \end{align}\tag{70}\] Since \(d\ge 5\), we can arrange that \(a^{-d/2+1}\ll a^{-1}\). Therefore, taking \(a\) big enough we can absorb the last term in the right-hand side of (70 ) to obtain \[\label{eq:54615} a^{-1}\|\chi_af\|+\|(i\nabla+b)\chi_af\|\lesssim a\|\langle x\rangle^{1+\epsilon}g\| +a^{2+3\epsilon-\rho}\sum_{j=0}^1\left\|\langle x\rangle^{-1-\epsilon}\nabla^j((1-\widetilde{\chi}_a)f)\right\|.\tag{71}\] Observe now that the identity (62 ) still holds with \([P,\chi_a]\) replaced by \(1-\widetilde{\chi}_a\). Using this together with Lemma 8, we get \[\begin{align} &\sum_{j=0}^1\left\|\langle x\rangle^{-1-\epsilon}\nabla^j((1-\widetilde{\chi}_a)f)\right\|\lesssim\sum_{j=0}^1\left\| \langle x\rangle^{-1-\epsilon}\nabla^j(P_0-\lambda^2\pm i\varepsilon)^{-1}\langle x\rangle^{-1-\epsilon}\right\| \|\langle x\rangle^{1+\epsilon}g\|\\ &+a^{2+2\epsilon-\rho}\sum_{j=0}^1\sum_{\ell_1+\ell_2\le 1}\left\| \langle x\rangle^{-1-\epsilon}\nabla^j(P_0-\lambda^2\pm i\varepsilon)^{-1}\nabla^{\ell_1}\langle x\rangle^{-1-\epsilon} \right\|\left\|\langle x\rangle^{-1-\epsilon}\nabla^{\ell_2}((1-\widetilde{\chi}_a)f)\right\|\\ &+\sum_{j=0}^1\sum_{\ell_1+\ell_2\le 1}\left\| \langle x\rangle^{-1-\epsilon}\nabla^j(P_0-\lambda^2\pm i\varepsilon)^{-1}\nabla^{\ell_1}\langle x\rangle^{-1-\epsilon} \right\|\left\|\nabla^{\ell_2}(\widetilde{\chi}_af)\right\|\\ &\lesssim\|\langle x\rangle^{1+\epsilon}g\|+a^{2+2\epsilon-\rho}\sum_{\ell=0}^1 \left\|\langle x\rangle^{-1-\epsilon}\nabla^{\ell}((1-\widetilde{\chi}_a)f)\right\| +\sum_{\ell=0}^1\left\|\nabla^{\ell}(\widetilde{\chi}_af)\right\|. \end{align}\] Taking \(a\) larger as needed we can absorb the second term in the right-hand side of the above inequality to obtain \[\label{eq:54616} \sum_{j=0}^1\left\|\langle x\rangle^{-1-\epsilon}\nabla^j((1-\widetilde{\chi}_a)f)\right\|\lesssim \|\langle x\rangle^{1+\epsilon}g\|+ \sum_{\ell=0}^1\left\|\nabla^{\ell}(\widetilde{\chi}_af)\right\|.\tag{72}\] By (71 ) and (72 ), \[\label{eq:54617} a^{-1}\|\chi_af\|+\|(i\nabla+b)\chi_af\|\lesssim a\|\langle x\rangle^{1+\epsilon}g\|+ a^{2+3\epsilon-\rho}\sum_{\ell=0}^1\left\|\nabla^{\ell}(\chi_af)\right\|.\tag{73}\] If \(\epsilon\) is small enough we have \(a^{2+3\epsilon-\rho}\ll a^{-1}\). Therefore, taking \(a\) even bigger if necessary we can absorb the last term in the right-hand side of (73 ) to obtain \[\label{eq:54618} a^{-1}\|\chi_af\|+\|(i\nabla+b)\chi_af\|\lesssim a\|\langle x\rangle^{1+\epsilon}g\|.\tag{74}\] With \(a\) now fixed we combine (72 ) and (74 ) to conclude \[\label{eq:54619} \sum_{j=0}^1\left\|\langle x\rangle^{-1-\epsilon}\nabla^jf\right\|\lesssim \|\langle x\rangle^{1+\epsilon}g\|,\tag{75}\] which clearly implies (?? ).  \(\Box\)

Let now \(\mathcal{O}\subset\mathbb{R}^d\) be a bounded domain with smooth boundary such that \(\Omega=\mathbb{R}^d\setminus\mathcal{O}\) is connected. In what follows in this section we will prove the following

Proof. We will adapt the proof of Theorem 11 to this case and will keep the same notations. In this follows \(\|\cdot\|\) and \(\langle\cdot,\cdot\rangle\) will denote the norm and the scalar product on \(L^2(\Omega)\). We take the parameter \(a\) big enough so that \(\chi_a=1\) on \(\mathcal{O}\). In this case the function \[f=(P-\lambda^2\pm i\varepsilon)^{-1}g\] satisfies the equation \[(P-\lambda^2\pm i\varepsilon)(\chi_af)=\chi_ag-[\Delta,\chi_a]f\] in \(\Omega\) and \(\chi_a f|_{\partial\Omega}=0\). Hence, by the Green formula, \[\begin{align} {\rm Re}\,\left\langle \chi_ag-[\Delta,\chi_a]f,\chi_af\right\rangle &={\rm Re}\,\left\langle P\chi_af,\chi_af\right\rangle -\lambda^2\|\chi_af\|^2\\ &=\|\nabla(\chi_af)\|^2+\left\langle V\chi_af,\chi_af\right\rangle-\lambda^2\|\chi_af\|^2\\ &\ge\|\nabla(\chi_af)\|^2_{L^2}-\lambda^2\|\chi_af\|^2. \end{align}\] Thus we obtain the inequality \[\label{eq:54620} \|\nabla(\chi_af)\|\le (\lambda+\gamma)\|\chi_af\|+\gamma^{-1}\|\chi_ag\|+\gamma^{-1}\|[\Delta,\chi_a]f\|\tag{76}\] for every \(\gamma>0\). On the other hand, by the Poincaré inequality (?? ), we have \[\label{eq:54621} \|\chi_af\|\le Ca\|\nabla(\chi_af)\|.\tag{77}\] Choosing \(\gamma=a^{-1}\gamma_0\) with \(\gamma_0>0\) small enough independent of \(a\) and \(\lambda\) small enough, we obtain from the above inequalities the estimate \[\label{eq:54622} a^{-1}\|\chi_af\|+\|\nabla(\chi_af)\|\le Ca\|\chi_ag\|+Ca\|[\Delta,\chi_a]f\|.\tag{78}\] On the other hand, by the resolvent identity (46 ) we have \[\label{eq:54623} (1-\eta)f=(P_0-\lambda^2\pm i\varepsilon)^{-1}(1-\eta)g+(P_0-\lambda^2\pm i\varepsilon)^{-1}([\Delta,\eta]-(1-\eta)V)f.\tag{79}\] Hence, if \(a\) is big enough, we have \[\label{eq:54624} [\Delta,\chi_a]f=[\Delta,\chi_a](P_0-\lambda^2\pm i\varepsilon)^{-1}(1-\eta)g+ [\Delta,\chi_a](P_0-\lambda^2\pm i\varepsilon)^{-1}([\Delta,\eta]-(1-\eta)V)f,\tag{80}\] \[\label{eq:54625} (1-\widetilde{\chi}_a)f=(1-\widetilde{\chi}_a)(P_0-\lambda^2\pm i\varepsilon)^{-1}(1-\eta)g+(1-\widetilde{\chi}_a)(P_0-\lambda^2\pm i\varepsilon)^{-1}([\Delta,\eta]-(1-\eta)V)f.\tag{81}\] With these formulas in hands, the proof now is exactly the same as the proof of Theorem 11. Therefore we omit the details.  \(\Box\)

6 Time decay estimates↩︎

In this section we use Theorems 8 and 10 to prove Theorem 1 for our self-adjoint operator \[P = (i\nabla + b)^2 + V : L^2(\Omega) \to L^2(\Omega).\] Recall that we consider the two cases a) and b) as described in Section 1. We will treat these cases separately as necessary. Throughout, we suppose \(P \ge 0\), for which \(V \ge 0\) suffices. Furthermore, we assume \(b\) and \(V\) obey (3 ) and that (4 ) holds for our domain \(\Omega\).

Given any integer \(m\ge 1\) there is a real-valued function \(\rho_m\in C_0^\infty({\mathbb{R}})\), \(\rho_m\ge 0\), such that \(\rho_m(\sigma)=0\) for \(\sigma\le 1\) and \(\sigma\ge 2\), \(\int_{-\infty}^\infty\rho_m(\sigma)d\sigma=1\), and \[\label{eq:6461} \left|\partial_\sigma^k\rho_m(\sigma)\right|\le C^{k+1}k!,\quad\forall\sigma\in {\mathbb{R}},\tag{82}\] for all integers \(0\le k\le m\) with a constant \(C>0\) independent of \(k\) and \(m\). Given any \(\delta>0\), set \[\psi_m(\lambda)=\int_{-\infty}^{\lambda/\delta} \rho_m(\sigma)d\sigma,\] so we have \(\partial_\lambda\psi_m(\lambda)=\delta^{-1}\rho_m(\lambda/\delta)\). Therefore, by (82 ), \[\label{eq:6462} \left|\partial_\lambda^k\psi_m(\lambda)\right|\le (C/\delta)^{k}k!,\quad\forall\lambda\in {\mathbb{R}},\tag{83}\] for all integers \(0\le k\le m\). Clearly, we also have \(0\le \psi_m(\lambda)\le 1\), and \(\psi_m(\lambda)=0\) for \(\lambda\le\delta\), \(\psi_m(\lambda)=1\) for \(\lambda\ge 2\delta\). Define the function \(\widetilde{\Psi}_m(\lambda,\lambda')\), \(\lambda,\lambda'\in [0, \infty)\), by \[\widetilde{\Psi}_m(\lambda,\lambda')= \begin{cases} \frac{\psi_m(\lambda)-\psi_m(\lambda')}{\lambda-\lambda'} & \lambda \neq\lambda', \\ \partial_\lambda\psi_m(\lambda) & \lambda = \lambda'. \end{cases}\] We note that an equivalent way to define \(\widetilde{\Psi}_m\) is \[\widetilde{\Psi}_m(\lambda,\lambda') = \delta^{-1}\int_0^1\rho_m(\lambda'(1-\sigma)/\delta+\lambda \sigma/\delta)d\sigma,\] which follows from \[\psi_m(\lambda)-\psi_m(\lambda')=\int_{\lambda'}^{\lambda} \partial_\tau (\psi_m(\tau))d\tau = \delta^{-1} \int_{\lambda'}^{\lambda} \rho_m(\tau) d \tau\] followed by the substitution \(\tau=(1-\sigma)\lambda'+\sigma \lambda\).

Set \[\Psi_m(\lambda,\lambda')=(\lambda+\lambda')^{-1}\widetilde{\Psi}_m(\lambda,\lambda'), \qquad \lambda, \, \lambda' \in [0, \infty).\] which is well-defined since \(\widetilde{\Psi}(\lambda, \lambda') = 0\) if \(\lambda, \, \lambda' \le \delta\). We need the following

Next we extend the function \(\psi_m(\lambda)\) to a smooth, even function on the whole of \({\mathbb{R}}\) (recall that \(\psi_m(\lambda) = 0\) for \(\lambda \le \delta\)) From now on our use of the notation \(\psi_m\) is meant to refer to this extension. We then smoothly extend \(\Psi(\lambda, \lambda') = (\lambda + \lambda')^{-1} \widetilde{\Psi}_m(\lambda, \lambda')\) to all of \({\mathbb{R}}_\lambda \times {\mathbb{R}}_{\lambda'}\) by \[\Psi_m(\lambda, \lambda') = \begin{cases} (\lambda^2 - (\lambda')^2)^{-1} (\psi_m(\lambda) - \psi_m(\lambda'))& \lambda \neq \lambda', \\ (2 \lambda)^{-1} \partial_\lambda \psi_m(\lambda) & \lambda = \lambda'. \end{cases}\] Observe the smoothness of this extension is justified by noticing that \(\Psi_m(\lambda, \lambda') = 0\) for \(|\lambda|,\, |\lambda'| \le \delta\), while near the set \(\{(\lambda, \lambda') : 0 \neq \lambda = \lambda'\}\), \[\Psi_m(\lambda, \lambda') = (\lambda + \lambda')^{-1}\int_0^1 (\partial \psi_m)((1 - \sigma)\lambda' + \sigma \lambda)d\sigma.\] We also have \(\Psi_m(\lambda, \lambda') = \Psi_m(|\lambda|, |\lambda'|)\).

Rearranging the expression for \(\Psi_m(\lambda, \lambda')\) yields \[\psi_m(\lambda') - \psi_m(\lambda) = ((\lambda')^2 - \lambda^2) \Psi_m(\lambda, \lambda'), \qquad \lambda, \, \lambda' \in {\mathbb{R}},\] whence for all \(\lambda \in {\mathbb{R}}\), \[\psi_m(P^{1/2})-\psi_m(\lambda)=(P-\lambda^2)\Psi_m(\lambda,P^{1/2}), \qquad \text{on } D(P).\] This identity will be used at a later stage in the analysis.

Using Lemma 3 we will prove

We will also need the following

7 Proof of Lemma 6↩︎

For \(0\le \lambda \le\delta/2\), \[\Psi_m(\lambda,P^{1/2})=\psi_m(P^{1/2})(P-\lambda^2)^{-1}\] and \(|x-\lambda|\ge\delta/2\) if \(x\in{\rm supp}\,\psi_m\). Hence in this case we have \[\begin{align} &\left\|\mu\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}\Psi_m(\lambda,P^{1/2})\mu\right\|\\ &\lesssim\sum_{j=0}^1\left\|P^{j/2}(P-(\lambda-i\varepsilon)^2)^{-1}\psi_m(P^{1/2})(P-\lambda^2)^{-1}\right\|\\ &\lesssim \sup_{x\in{\rm supp}\,\psi_m}(|x|+1)|x^2-(\lambda-i\varepsilon)^2|^{-1}|x^2-\lambda^2|^{-1}\lesssim 1 \end{align}\] uniformly in \(\varepsilon\) and \(\lambda\). Note that to get the second inequality we used ?? Let now \(\lambda \ge\delta/2\). Let \(\chi_1,\, \chi_2,\, \chi_3 \in C^\infty({\mathbb{R}}^+)\) be such that \(\chi_1+\chi_2+\chi_3\equiv 1\) on \({\mathbb{R}}^+\), \(\chi_1(\lambda')=1\) for \(\lambda'\le \delta/3\), \(\chi_1(\lambda')=0\) for \(\lambda'\ge \delta/2\), \(\chi_3(\lambda')=0\) for \(\lambda'\le 3\delta\), \(\chi_3(\lambda')=1\) for \(\lambda'\ge 4\delta\). Then \[\chi_1(P^{1/2})\Psi_m(\lambda,P^{1/2})=-\psi_m(\lambda)\chi_1(P^{1/2})(P-\lambda^2)^{-1}\] and \(|x-\lambda|\ge\delta/2\) if \(x\in{\rm supp}\,\chi_1\) and \(\lambda\in{\rm supp}\,\psi_m\). Hence \[\begin{align} &\left\|\mu\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}\chi_1(P^{1/2})\Psi_m(\lambda,P^{1/2})\mu\right\|\\ &\lesssim\sum_{j=0}^1\left\|P^{j/2}(P-(\lambda-i\varepsilon)^2)^{-1} \psi_m(\lambda) \chi_1(P^{1/2})(P-\lambda^2)^{-1}\right\|\\ &\lesssim \sup_{x\in{\rm supp}\,\chi_1,\,\lambda\in{\rm supp}\,\psi_m}(|x|+1)|x^2-(\lambda-i\varepsilon)^2|^{-1}|x^2-\lambda^2|^{-1}\lesssim 1 \end{align}\] uniformly in \(\varepsilon\) and \(\lambda\). Furthermore, we have \[\chi_3(P^{1/2})\Psi_m(\lambda,P^{1/2})=(1-\psi_m)(\lambda)\chi_3(P^{1/2})(P-\lambda^2)^{-1}\] and \(|x-\lambda|\ge\delta\) if \(x\in{\rm supp}\,\chi_3\) and \(\lambda\in{\rm supp}\,(1-\psi_m)\). In the same way as above, we get \[\left\|\mu\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}\chi_3(P^{1/2})\Psi_m(\lambda,P^{1/2})\mu\right\|\lesssim 1\] uniformly in \(\varepsilon\) and \(\lambda\). It remains to show that \[\label{eq:7461} \left\|\mu\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}\chi_2(P^{1/2})\Psi_m(\lambda,P^{1/2})\mu\right\|\lesssim 1\tag{108}\] uniformly in \(\varepsilon\) and \(\lambda\). Clearly, the function \[\varphi(x)=\chi_2(x^{1/2})\Psi_m(\lambda,x^{1/2})\] belongs to \(C_0^\infty({\mathbb{R}})\) and \(|\partial_x^n\varphi(x)|\lesssim 1\) for \(n\le 2\). Then there is an almost analytic extension, \(\widetilde{\varphi}\), of \(\varphi\) on \({\mathbb{C}}\) such that \(\widetilde{\varphi}|_{{\mathbb{R}}}=\varphi\) and \[\label{eq:7462} \left|\overline{\partial}\widetilde{\varphi}(z)\right|\le C_N|{\rm Im}\,z|^N\sum_{n=0}^N\sup_x|\partial_x^n\varphi(x)|,\quad\forall N\ge 0,\tag{109}\] where the constant \(C_N\) does not depend on the function \(\varphi\); \(\widetilde{\varphi}\) is supported in a complex neighbourhood of supp\(\,\varphi\). In our case \(\widetilde{\varphi}\) is supported in a complex neighbourhood of supp\(\,\chi_2(x^{1/2})\). We are going to use the Helffer-Sjöstrand formula \[\label{eq:7463} \varphi(P)=\frac{1}{\pi}\int\overline{\partial}\widetilde{\varphi}(z)(P-z)^{-1}dx dy,\quad z=x+iy.\tag{110}\] Thus the operator in (108 ) can be written in the form \[\label{eq:7464} \frac{1}{\pi}\int\overline{\partial}\widetilde{\varphi}(z) \mu\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}(P-z)^{-1}\mu dx dy.\tag{111}\] From the resolvent identity \[\begin{align} &(\lambda^2 -\varepsilon^2-z)(P-(\lambda-i\varepsilon)^2)^{-1}(P-z)^{-1}\\ &=2i\varepsilon\lambda(P-(\lambda-i\varepsilon)^2)^{-1}(P-z)^{-1}\\ &+(P-(\lambda-i\varepsilon)^2)^{-1}-(P-z)^{-1}, \end{align}\] we get \[\begin{align} &|\lambda^2-\varepsilon^2-z|\left\|\mu\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}(P-z)^{-1}\mu\right\|\\ &\le 2\varepsilon|\lambda|\left\|\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}\right\|\left\|(P-z)^{-1}\right\|\\ &+\left\|\mu\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}\mu\right\|+ \left\|(P-z)^{-1}\right\|\\ &\lesssim|{\rm Im}\,z|^{-1}\varepsilon|\lambda|\sum_{j=0}^1\left\|P^{j/2}(P-(\lambda-i\varepsilon)^2)^{-1}\right\|\\ &+ 1+|{\rm Im}\,z|^{-1}, \end{align}\] where we have used the resolvent estimate (?? ) in the case a) and (?? ) in the case b). Moreover, \[\label{eq:7465} \varepsilon |\lambda| \|(P-(\lambda-i\varepsilon)^2)^{-1}\| \lesssim 1,\tag{112}\] and \[\label{eq:7466} \varepsilon |\lambda| \|P^{1/2}(P-(\lambda-i\varepsilon)^2)^{-1}\| \lesssim |\lambda| + 1.\tag{113}\] Indeed, 113 is a consequence of 112 , the identity \[P(P - (\lambda - i\varepsilon)^2)^{-1} = I + (\lambda - i \varepsilon)^2 (P - (\lambda - i\varepsilon)^2)^{-1}\] and the estimate \[\|P^{1/2}u\|^2_{L^2} = \langle Pu, u \rangle_{L^2} \le \|Pu\|_{L^2} \|u\|_{L^2}, \qquad u \in D(P).\] Combining the previous inequalities implies \[|\lambda^2-\varepsilon^2-z|\left\|\mu\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}(P-z)^{-1}\mu\right\| \lesssim |\lambda| + 1 + |{\rm Im}\,z|^{-1}.\] On the other hand, \[|\lambda^2-\varepsilon^2-z|=((\lambda^2 - \varepsilon^2 - {\rm Re}\,z)^2 + |{\rm Im}\,z|^2)^{1/2}\ge |{\rm Im}\,z|,\] while for large \(|\lambda|\) and \(z\in{\rm supp}\,\widetilde{\varphi}\) we have \[|\lambda^2-\varepsilon^2-z|\ge |\lambda|^2/2.\] Therefore, we have on the support of \(\widetilde{\varphi}\), \[\label{eq:7467} \left\|\mu\nabla^{\ell}(P-(\lambda-i\varepsilon)^2)^{-1}(P-z)^{-1}\mu\right\|\lesssim |{\rm Im}\,z|^{-2}\tag{114}\] uniformly in \(\varepsilon\) and \(\lambda\). It follows from (114 ) together with the formula (110 ) and (109 ) with \(N=2\) that the operator (111 ) is bounded uniformly in \(\varepsilon\) and \(\lambda\), which in turn implies (108 ).

8 Self-adjointness of the magnetic Schrödinger operator on \(L^2({\mathbb{R}}^d)\)↩︎

In this appendix we discuss self-adjointness of 1 when \(\Omega = {\mathbb{R}}^d\), \(b \in L^\infty({\mathbb{R}}^d ; {\mathbb{R}}^d)\) is not identically zero, and \(V \in L^\infty({\mathbb{R}}^d ; {\mathbb{R}})\). In this case the self-adjoint realization of 1 we use throughout the paper is constructed via a sesquilinear form as follows. On \(H^1({\mathbb{R}}^d) \times H^1({\mathbb{R}}^d)\) put \[\label{form32for32mag32Schro} q (u, v) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =\int_{{\mathbb{R}}^d} \nabla \overline{u} \cdot \nabla v - i \overline{u}b \cdot \nabla v + i vb \cdot \nabla \overline{u} + (V + |b|^2)\overline{u} v dx.\tag{115}\] For any \(\epsilon \ge 0\), by Cauchy-Schwarz and Young’s inequality, \[\begin{align} \big|\int_{{\mathbb{R}}^d} i ub \cdot \nabla \overline{u} - i\overline{u}b \cdot \nabla u dx\big| & = 2 \big| {\rm Im}\:\int_{{\mathbb{R}}^d} \overline{u}b \cdot \nabla u dx\big| \\ & \le (1 - \epsilon) \| \nabla u \|^2_{L^2} + \frac{1}{1 - \epsilon} \| bu \|^2_{L^2}, \end{align}\] whence, \[\label{semiboundedness} q (u, u) + \| u\|_{L^2}^2 \ge \epsilon \| \nabla u \|^2_{L^2} + \int_{{\mathbb{R}}^d} \big(V + 1 - \frac{\epsilon}{1 - \epsilon} \| b \|^2_{L^\infty({\mathbb{R}}^d; {\mathbb{R}}^d)} \big) |u|^2 dx, \qquad \epsilon \ge 0.\tag{116}\] The last estimate shows that 115 is semibounded and closed in the sense of [13]. Moreover, if \(V \ge 0\), setting \(\epsilon = 0\) yields \(q(u,u) \ge 0\).

By [13], there exists a unique, densely defined self-adjoint operator \(P\) whose quadratic form domain is \(H^1({\mathbb{R}}^n)\), and whose associated sesquilinear form coincides with \(q\) on \(H^1({\mathbb{R}}^n)\). The domain of \(P\) is \[\begin{gather} D(P) = \{ u \in H^1({\mathbb{R}}^d) : \text{there is } \tilde{u} \in L^2({\mathbb{R}}^d) \text{ such that } q(u, v) = \langle \tilde{u}, v\rangle_{L^2} \text{ for all } v \in H^1({\mathbb{R}}^d)\}, \\ Pu = \tilde{u}. \end{gather}\] The equality \(q(u, v) = \langle \tilde{u}, v\rangle_{L^2}\) for \(u \in D(P)\) and \(v \in H^1({\mathbb{R}}^d)\) shows that, in the sense of distributions on \({\mathbb{R}}^d\), \[Pu = -\Delta u + i \nabla \cdot (u b) + ib \cdot \nabla u + (V + |b|^2)u.\] Here, if \((\cdot, \cdot)\) denotes distributional pairing, we define the divergence of a distribution \(u\) by \((\nabla \cdot u, v) \mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =-(u, \nabla \cdot v)\).

In Section 3 we make use of several mapping properties of \((P- z)^{-1}\) for \(z\) in the resolvent set \(\rho(P)\) of \(P\), which we now formulate. Recall that the negative index Sobolev space \(H^{-1}({\mathbb{R}}^d)\) is isometrically isomorphic to the dual space of \(H^1({\mathbb{R}}^d)\) under the mapping \(H^{-1}({\mathbb{R}}^d) \ni u \mapsto \langle u , \cdot \rangle_{L^2}\), and that \[\| v\|_{H^{-1}} = \sup_{0 \neq u \in H^1} \frac{ \langle v, u \rangle_{L^2}}{\| u\|_{H^1}}.\] We show that for any \(z \in \rho(P)\), \((P- z)^{-1}\) maps boundedly from \(H^{-1}({\mathbb{R}}^d)\) to \(H^1({\mathbb{R}}^d)\). In particular, there exists \(C_z > 0\) so that \[\label{H32minus32132to32H132bd} \|(P- z)^{-1} u\|_{H^1} \le C_z \| u\|_{H^{-1}}, \qquad u \in C^\infty_0({\mathbb{R}}^d).\tag{117}\] We reuse the constant \(C_z\) below. Its precise value changes from line to line, but it stays independent of \(u\).

By fixing \(0 < \epsilon \ll 1\) in 116 , we get \(C > 0\) independent of \(u \in L^2({\mathbb{R}}^d)\) such that \[\label{H132to32L232bd} \begin{align} \| (P - z)^{-1} u \|^2_{H^1} &\le C (\| (P - z)^{-1} u \|^2_{L^2} + q((P - z)^{-1} u , (P - z)^{-1} u ) \\ &= C (\| (P - z)^{-1} u \|^2_{L^2} + \langle P(P - z)^{-1} u , (P - z)^{-1} u \rangle_{L^2}) \\ &= C (\| (P - z)^{-1} u \|^2_{L^2} + \langle u , (P - z)^{-1} u \rangle_{L^2} + \overline{z} \| (P - z)^{-1} u\|^2_{L^2}). \end{align}\tag{118}\] Since \(\| (P - z)^{-1} u \|_{L^2} \le C_z \|u\|_{L^2}\) this implies \[\|(P- z)^{-1} u\|_{H^1} \le C_z \| u\|_{L^2}, \qquad u \in L^2({\mathbb{R}}^d).\] Thus, for any \(u \in C^\infty_0({\mathbb{R}}^d)\) and \(v \in L^2({\mathbb{R}}^d)\), \[|\langle (P- z)^{-1} u, v \rangle_{L^2}| = | \langle u, (P- \overline{z})^{-1}v \rangle_{L^2}| \le \|u \|_{H^{-1}} \|(P-z)^{-1}v \|_{H^1} \le C_z \| u \|_{H^{-1}} \| v \|_{L^2}.\] Therefore we conclude, \[\label{H32minus32132to32L232bd} \|(P- z)^{-1} u\|_{L^2} \le C_z \| u\|_{H^{-1}}, \qquad u \in C^\infty_0({\mathbb{R}}^d).\tag{119}\] Now 117 follows from 118 and 119 . Indeed, for \(u \in C^\infty_0({\mathbb{R}}^d)\),

\[\begin{align} \| (P &- z)^{-1} u \|^2_{H^1}\\ & \le C (\| (P - z)^{-1} u \|^2_{L^2} + \langle u , (P - z)^{-1} u \rangle_{L^2} + \overline{z} \|(P - z)^{-1} u\|^2_{L^2}) \\ & \le C_z (\| (P - z)^{-1} u \|_{L^2} + \| u \|_{H^{-1}}) \| (P-z)^{-1} u\|_{H^1} \\ & \le C_z \|u \|_{H^{-1}} \| (P-z)^{-1} u\|_{H^1} . \end{align}\]

We use 117 to verify a resolvent identity that we apply in Section 3. Let \(P_0 = - \Delta\) denote the free Laplacian on \({\mathbb{R}}^d\). We show that for \(u \in H^2({\mathbb{R}}^d)\) and \(z \in \rho(P)\) \[\label{key32resolv32id32for32mag32Schro} (P - z)^{-1} (\widetilde{V} + i\nabla \cdot b + ib \cdot \nabla ) u = u - (P - z)^{-1}(P_0 - z)u,\tag{120}\] where \(\widetilde{V} = V + |b|^2\) and we note that the divergence is a bounded operator \(L^2({\mathbb{R}}^d; {\mathbb{C}}^d) \to H^{-1}({\mathbb{R}}^d)\). To show 120 , let \(u_k \in C^\infty_0({\mathbb{R}}^d ; {\mathbb{C}}^d)\) converge to \(bu\) in \(L^2({\mathbb{R}}^d; {\mathbb{C}}^d)\), in which case by 117 we have that \((P - z)^{-1} i\nabla \cdot u_{k}\) converges to \((P - z)^{-1} i\nabla \cdot bu\) in \(L^2({\mathbb{R}}^d)\). Then for any \(v \in L^2({\mathbb{R}}^d)\), \[\begin{align} \langle v, &(P - z)^{-1} (\widetilde{V} + i\nabla \cdot b + ib \cdot \nabla ) u \rangle_{L^2} \\ &= \langle -ib(P- \overline{z})^{-1} v, \nabla u \rangle_{L^2({\mathbb{R}}^d; {\mathbb{C}}^d)}+ \langle \widetilde{V} (P- \overline{z})^{-1} v, u \rangle_{L^2} + \lim_{k \to \infty} \langle v, (P - z)^{-1} i \nabla \cdot u_k \rangle_{L^2} \\ &= \langle (P- \overline{z})^{-1} v, -\Delta u \rangle_{L^2} - \langle ib(P- \overline{z})^{-1} v, \nabla u \rangle_{L^2({\mathbb{R}}^d; {\mathbb{C}}^d)}+ \langle ( \widetilde{V} + i b \cdot \nabla - \overline{z}) (P- \overline{z})^{-1} v, u \rangle_{L^2} \\ &- \langle v, (P- z)^{-1} (P_0 -z) u \rangle_{L^2} \end{align}\] Now 120 follows since \((-\Delta + i \nabla \cdot b + ib \cdot \nabla + \widetilde{V} - \overline{z}) (P- \overline{z})^{-1} v = v\) in the sense of distributions.

9 Analytic functions in a strip↩︎

Let the function \(f(\lambda)\) be analytic in \(\{\lambda\in\mathbb{C}: A<{\rm Re}\,\lambda,\,|{\rm Im}\,\lambda|<\gamma\}\), where \(\gamma>0\) is some constant, while \(A\) is either a constant or \(A=-\infty\). Let also \(f\) satisfy in this region the bound \[\label{eq:B461} |f(\lambda)|\le M\tag{121}\] with some constant \(M>0\). Let \(\gamma_1\) be any constant such that \(0<\gamma_1<\gamma\). If \(A\) is a constant we take any constant \(A_1\) such that \(A_1>A\). If \(A=-\infty\) we take \(A_1=-\infty\). For such functions we will prove the following

Proof. The bound (?? ) follows from (121 ) and the Cauchy formula \[\label{eq:B464} \partial_\lambda^kf(\lambda)=\frac{k!}{2\pi i}\int_{|z-\lambda|=\sigma}\frac{f(z)}{(z-\lambda)^{k+1}}dz\tag{122}\] for every integer \(k\ge 0\), where \(\sigma\) is a constant such that \(0<\sigma<\gamma-\gamma_1\). If \(A\) and \(A_1\) are constants we also require that \(\sigma<A_1-A\). Furthermore, we have \[f(\lambda)-f({\rm Re}\,\lambda)=i{\rm Im}\,\lambda f'({\rm Re}\,\lambda+it)\] with some real \(t\) such that \(|t|\le|{\rm Im}\,\lambda|\), where \(f'\) denotes the first derivative of \(f\). Therefore, (?? ) follows from (?? ) with \(k=1\).  \(\Box\)

10 Resolvent bounds for the free resolvent↩︎

The following estimates for the free resolvent are well-known and therefore we omit the proof.

Next lemma is well-known when the function \(\mu\) is compactly supported. We show that it still holds with \(\mu=e^{-c\langle x\rangle/2}\), \(c>0\).

Proof. Note first that (?? ) follows from (?? ) and (?? ). Since the operator \(\partial_x^\beta\) commutes with the free resolvent, it suffices to prove the lemma with \(\beta=0\) and \(|\alpha|\le 2\).

It is well-known that the kernel \(K(x,y;\lambda)\) of the free resolvent \[R_0(\lambda)=(P_0-\lambda^2)^{-1},\quad {\rm Im}\,\lambda<0,\] can be expressed in terms of the Hankel functions by the formula \[K(x,y;\lambda)=i2^{-2}(2\pi)^{-\frac{d-2}{2}}\lambda^{\frac{d-2}{2}}|x-y|^{-\frac{d-2}{2}}H^-_{\frac{d-2}{2}}(\lambda|x-y|).\] It is also well-known that \(z^{\frac{d-2}{2}}H^-_{\frac{d-2}{2}}(z)\) extends analytically from \(\mathbb{C}^-\) to the complex plane \(\mathbb{C}\) if \(d\) is odd and to the Riemann surface of the logarithm if \(d\) is even and satisfies the bounds \[\label{eq:C467} \left|H^-_{\frac{d-2}{2}}(z)\right|\lesssim \left\{ \begin{array}{lll} |\log|z||+1&for& |z|\le 1,\,d=2,\\ |z|^{-\frac{d-2}{2}}&for& |z|\le 1,\,d\ge 3,\\ |z|^{-1/2}e^{{\rm Im}\,z}&for& |z|\ge 1,\,d\ge 2. \end{array} \right.\tag{123}\] Hence the kernel \(K\) extends analytically in \(\lambda\) from \(\mathbb{C}^-\) to the complex plane \(\mathbb{C}\) if \(d\) is odd and to the Riemann surface of the logarithm if \(d\) is even and satisfies the bound \[\label{eq:C468} |K(x,y;\lambda)|\lesssim G_d(|x-y|)+|\lambda|^{(d-3)/2}|x-y|^{-(d-1)/2}e^{{\rm Im}\,\lambda|x-y|},\tag{124}\] where \[G_d(\sigma)= \left\{ \begin{array}{lll} |\log\sigma|+|\log|\lambda||+1&if& d=2,\\ \sigma^{-d+2}&if& d\ge 3. \end{array} \right.\] Fix a constant \(0<\gamma_0<c/2\). Since \[\mu(x)\mu(y)\le e^{-c|x-y|/2},\] we deduce from (124 ), \[\label{eq:C469} |\mu(x)K(x,y;\lambda)\mu(y)|\lesssim \left(G_d(|x-y|) +|\lambda|^{(d-3)/2}|x-y|^{-(d-1)/2}\right)e^{-(c/2-\gamma_0)|x-y|}\tag{125}\] for \({\rm Im}\,\lambda\le \gamma_0\). It follows from (125 ) and Schur’s lemma that the operator \(\mu R_0(\lambda)\mu\), is bounded on \(L^2\) for \({\rm Im}\,\lambda\le\gamma_0\) with norm \(O\left(1+|\lambda|^{(d-3)/2}\right)\). Therefore the operator \(\mu R_0(\lambda)\mu: L^2\to L^2\) extends analytically from \(\mathbb{C}^-\) to \(\mathcal{L}_{\gamma_0}\). This also implies the bound (?? ) with \(\alpha=\beta=0\) for \(|\lambda|\le 1\) if \(d\ge 3\).

To prove the bound (?? ) in the other cases we will follow [15] where (?? ) with \(\alpha=\beta=0\) is proved for compactly supported \(\mu\) (see Proposition 2.1 of [15]). When \(\mu\) decays exponentially the proof is the same but we will sketch the main points for the sake of completeness. It is based on the formula \[\label{eq:C4610} K(x,y;\lambda)-K(x,y;-\lambda)=i2^{-1}(2\pi)^{1-d}\lambda^{d-2}\int_{\mathbb{S}^{d-1}}e^{i\lambda\langle x-y,w\rangle}dw,\tag{126}\] where \(\mathbb{S}^{d-1}\) denotes the unit sphere in \(\mathbb{R}^d\). From (126 ) we get the formula \[\label{eq:C4611} \mu\partial_x^\alpha R_0(\lambda)\mu-\mu\partial_x^\alpha R_0(-\lambda)\mu =i2^{-1}(2\pi)^{1-d}\lambda^{d-2+|\alpha|}\mathcal{A}_\mu^{(\alpha)}(\lambda) \mathcal{A}_\mu^{(0)}(\overline{\lambda})^*,\tag{127}\] for any muli-index \(\alpha\), where \[\mathcal{A}_\mu^{(\alpha)}(\lambda):L^2(\mathbb{S}^{d-1})\to L^2(\mathbb{R}^d)\] is the operator with kernel \[\begin{align} A_\mu^{(\alpha)}(x,w)=i^{|\alpha|}w^{\alpha}\mu(x)e^{i\lambda\langle x,w\rangle},\quad x\in\mathbb{R}^d,\,w\in \mathbb{S}^{d-1}. \end{align}\] Our goal is to prove the bound \[\label{eq:C4612} \left\|\mu\partial_x^\alpha R_0(\lambda)\mu-\mu\partial_x^\alpha R_0(-\lambda)\mu \right\|\lesssim |\lambda|^{-1+|\alpha|}\tag{128}\] for all \(\lambda\in\mathbb{C}\), \(\lambda\neq 0\), such that \(|{\rm Im}\,\lambda|\le\gamma_0\). In view of (127 ), it suffices to prove the bound \[\label{eq:C4613} \left\|\mathcal{A}_\mu^{(\alpha)}(\lambda)\right\|_{L^2(\mathbb{S}^{d-1})\to L^2(\mathbb{R}^d)}\lesssim |\lambda|^{-\frac{d-1}{2}}.\tag{129}\] On the other hand, in view of Plancherel’s identity, the norm in (129 ) is equivalent to the norm of the operator \[\mathcal{F}\mathcal{A}_\mu^{(\alpha)}(\lambda):L^2(\mathbb{S}^{d-1})\to L^2(\mathbb{R}^d),\] where \(\mathcal{F}\) is the Fourier transform. Since the kernel of this operator is equal to \(i^{|\alpha|}w^{\alpha}(\mathcal{F}\mu)(\xi-i\lambda w)\), by Schur’s lemma it suffices to show that \[\label{eq:C4614} \int_{\mathbb{R}^{d}}|(\mathcal{F}\mu)(\xi-i\lambda w)|d\xi\lesssim 1,\quad \int_{\mathbb{S}^{d-1}}|(\mathcal{F}\mu)(\xi-i\lambda w)|dw\lesssim |\lambda|^{-d+1}\tag{130}\] for all \(\lambda\in\mathbb{C}\), \(\lambda\neq 0\), such that \(|{\rm Im}\,\lambda|\le\gamma_0\). To this end, we will use that \((\mathcal{F}\mu)(\xi)\) extends to all \(\xi\in\mathbb{C}^d\) such that \(|{\rm Im}\,\xi|\le\gamma_0\) and satisfies the bounds \[\left|\xi^\beta(\mathcal{F}\mu)(\xi)\right|\lesssim \int_{\mathbb{R}^{d}}\left|\partial_x^\beta\mu(x)\right|e^{|{\rm Im}\,\xi||x|}dx \lesssim \int_{\mathbb{R}^{d}}\mu(x)e^{|{\rm Im}\,\xi||x|}dx\lesssim\int_{\mathbb{R}^{d}}e^{-(c/2-\gamma_0)|x|}dx\lesssim 1\] for all multi-indices \(\beta\). Thus we obtain that given any integer \(M\ge 0\) there is a constant \(C_M>0\) such that \[\label{eq:C4615} \left|(\mathcal{F}\mu)(\xi)\right|\le C_M(|\xi|+1)^{-M}\tag{131}\] for all \(\xi\in\mathbb{C}^d\) such that \(|{\rm Im}\,\xi|\le\gamma_0\). We now apply (131 ) with \(\xi-i\lambda w\), \(\xi\in\mathbb{R}^d\). Then the bounds (130 ) follow from (131 ) in the same way as in Section 2 of [15].

It is easy to see now that the estimate (128 ) implies (?? ). Indeed, since the bound (?? ) is trivial on \({\rm Im}\,\lambda=-\gamma_0\), by (128 ) with \(|\alpha|\le 2\) we conclude that it also holds on \({\rm Im}\,\lambda=\gamma_0\). Then the Phragmén-Lindelöf principle implies that (?? ) holds for \(|{\rm Im}\,\lambda|\le\gamma_0\).  \(\Box\)

11 Poincaré inequality↩︎

Proof. Clearly, it suffices to prove (?? ) for all functions \(f\in C_0^1(\mathbb{R}^d)\). Let \((r,w)\in (0,\infty)\times \mathbb{S}^{d-1}\) be the polar coordinates and set \[u(r,w)=f(rw)e^{-i\int_0^rw\cdot b(\sigma w)d\sigma}.\] We have \[\begin{align} \int_0^\infty r^{d-3}|u(r,w)|^2dr&=(d-2)^{-1}\int_0^\infty |u(r,w)|^2 (r^{d-2})'dr\\ &=-2(d-2)^{-1}{\rm Re}\,\int_0^\infty u'(r,w) \overline{u(r,w)}r^{d-2}dr, \end{align}\] where the prime notation denotes the first derivative of a function with respect to \(r\). Hence \[\int_0^\infty r^{d-3}|u(r,w)|^2dr\lesssim \left(\int_0^\infty r^{d-1}|u'(r,w)|^2dr\right)^{1/2} \left(\int_0^\infty r^{d-3}|u(r,w)|^2dr\right)^{1/2}\] which implies \[\int_0^\infty r^{d-3}|u(r,w)|^2dr\lesssim \int_0^\infty r^{d-1}|u'(r,w)|^2dr.\] Integrating this inequlity with respect to \(w\) leads to the estimate \[\label{eq:D463} \left\|r^{-1}u\right\|_{L^2(\mathbb{R}^d)}\lesssim \left\|\partial_r u\right\|_{L^2(\mathbb{R}^d)}.\tag{132}\] Observe now that \[\label{eq:D464} \left|\partial_r u\right|=\left|(i\partial_r+w\cdot b(rw))f\right|=\left|w\cdot(i\nabla+ b(rw))f\right| \le \left|(i\nabla+ b)f\right|.\tag{133}\] Clearly, (?? ) follows from (132 ) and (133 ).

The inequality (?? ) follows from (?? ) in the following manner. Given \(f \in H^1(\Omega)\) with \(f = 0\) on \(\partial \Omega\), by [27], there exists a sequence \(f_k \in C^\infty_0(\Omega) \subseteq C^\infty_0({\mathbb{R}}^d)\) converging to \(f\) in \(H^1\)-norm. By taking a subsequence if necessary, which we still denote by \(f_k\), we can suppose the \(f_k\) converge pointwise almost everywhere to \(f\) with respect to the Lebesgue measure. Then by Fatou’s lemma and ?? \[\| |x|^{-1} f\|^2_{L^2(\Omega)} = \liminf_{k \to \infty} \||x|^{-1} f_k \|_{L^2(\Omega)} \lesssim \liminf_{k \to \infty} \| \nabla f_k \|_{L^2(\Omega)} = \| \nabla f \|_{L^2(\Omega)} .\]  \(\Box\)

References↩︎