Semiclassical Resolvent Research Articles

Semiclassical resolvent bounds for short range L ∞ potentials with singularities at the origin

We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity.

Sharp Resolvent Estimate for the Damped-Wave Baouendi–Grushin Operator and Applications

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$ with H\"older dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on $\mathbb{T}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.

Semiclassical resolvent bounds for compactly supported radial potentials

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator −h2Δ+V(|x|)−E in dimension n≥2, where h,E>0, and V:[0,∞)→R is L∞ and compactly supported. The weighted resolvent norm grows no faster than exp⁡(Ch−1), while an exterior weighted norm grows ∼h−1. We introduce a new method based on the Mellin transform to handle the two-dimensional case.

The Semiclassical Resolvent on Conic Manifolds and Application to Schrödinger Equations

In this article, we shall construct the resolvent of the Laplacian at high energies near the spectrum on non-product conic manifolds with a single cone tip. Microlocally, the resolvent kernel is the sum of b-pseudodifferential operators, scattering pseudodifferential operators and intersecting Legendrian distributions. As an application, we shall establish Strichartz estimates for Schrödinger equations on non-compact manifolds with multiple non-product conic singularities.

Semiclassical resolvent bounds for weakly decaying potentials

In this note, we prove weighted resolvent estimates for the semiclassical Schrodinger operator $-h^2 \Delta + V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at infinity or to obey a radial $\alpha$-Holder continuity condition, $0\leq \alpha \leq 1$, with sufficient decay of the local radial $C^\alpha$ norm toward infinity. Note, however, that in the Holder case, the potential need \emph{not} decay. If the dimension $n \ge 3$, the resolvent bound is of the form $\exp \left(C h^{-1 - \frac{1 - \alpha}{3 + \alpha}} [(1-\alpha) \log(h^{-1})+c]\right)$, while for $n = 1$ it is of the form $\exp(Ch^{-1})$. A new type of weight and phase function construction allows us to reduce the necessary decay even in the pure $L^\infty$ case.

Improved resolvent bounds for radial potentials

We prove semiclassical resolvent estimates for the Schrödinger operator in \({\mathbb {R}}^d\), \(d\ge 3\), with real-valued radial potentials \(V\in L^\infty ({\mathbb {R}}^d)\). In particular, we show that if \(V(x)={{\mathcal {O}}}\left( \langle x\rangle ^{-\delta }\right) \) with \(\delta >2\), then the resolvent bound is of the form \(\exp \left( Ch^{-4/3}\right) \) with some constant \(C>0\). We also get resolvent bounds when \(1<\delta \le 2\). For slowly decaying \(\alpha \)—Hölder potentials, we get better resolvent bounds of the form \(\exp \left( Ch^{-4/(\alpha +3)}\right) \).

Semiclassical resolvent estimates for Hölder potentials

We first prove semiclassical resolvent estimates for the Schrodinger operator in R d , d ≥ 3, with real-valued potentials which are Holder with respect to the radial variable. Then we extend these resolvent estimates to exterior domains in R d , d ≥ 2, and real-valued potentials which are Holder with respect to the space variable. As an application, we obtain the rate of the decay of the local energy of the solutions to the wave equation with a refraction index which may be Holder, Lipschitz or just L ∞ .

Semiclassical resolvent bound for compactly supporte $L^\infty$ potentials

We give an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger operators in dimension n \ge 1 . We require the potential belong to L^\infty(\mathbb{R}^n) and have compact support, but do not require that it have distributional derivatives in L^\infty(\mathbb{R}^n) . The weighted resolvent norm is bounded by e^{Ch^{-4/3}\log(h^{-1})} , where h is the semiclassical parameter.

Semiclassical Resolvent Estimates for $$L^\infty $$ Potentials on Riemannian Manifolds

We prove semi-classical resolvent estimates for the Schr{\"o}dinger operator with a real-valued L $\infty$ potential on non-compact, connected Riemannian manifolds which may have a compact smooth boundary. We show that the resolvent bound depends on the structure of the man-ifold at infinity. In particular, we show that for compactly supported real-valued L $\infty$ potentials and asymptoticaly Euclidean manifolds the resolvent bound is of the form exp(Ch --4/3 log(h --1)), while for asymptoticaly hyperbolic manifolds it is of the form exp(Ch --4/3), where C > 0 is some constant.

Semiclassical Estimates for Scattering on the Real Line

We prove explicit semiclassical resolvent estimates for an integrable potential on the real line. The proof is a comparatively easy case of the spherical energies method, which has been used to prove similar theorems in higher dimensions and in more complicated geometric situations. The novelty in our results lies in the weakness of the assumptions on the potential.

Semiclassical resolvent bounds in dimension two

We give an elementary proof of weighted resolvent bounds for semiclassical Schrodinger operators in dimension two. We require the potential function to be Lipschitz with long range decay. The resolvent norm grows exponentially in the inverse semiclassical parameter, but near infinity it grows linearly. Our result covers the missing case from the work of Datchev.

Semiclassical resolvent estimates for short-range L ∞ potentials. II

We prove semiclassical resolvent estimates for real-valued potentials V ∈ L ∞ ( R n ) , n ⩾ 3 , of the form V = V L + V S , where V L is a long-range potential which is C 1 with respect to the radial variable, while V S is a short-range potential sat

Semiclassical resolvent estimates for bounded potentials

We study the cut-off resolvent of semiclassical Schr{\"o}dinger operators on $\mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $\lambda^2$ in a compact interval in $\mathbb{R}_+$ and for any smooth cut-off function $\chi$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has \begin{equation*} \| \chi (-h^2\Delta + V-\lambda^2)^{-1} \chi \|_{L^2\to H^1} \leq C \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This bound shows in particular an upper bound on the imaginary parts of the resonances $\lambda$, defined as a pole of the meromorphic continuation of the resolvent $(-h^2\Delta + V-\lambda^2)^{-1}$ as an operator $L^2_{\mathrm{comp}}\to H^2_{\mathrm{loc}}$: any resonance $\lambda$ with real part in a compact interval away from $0$ has imaginary part at most \begin{equation*} \mathrm{Im} \lambda \leq - C^{-1} \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This is related to a conjecture by Landis: The principal Carleman estimate in our proof provides as well a lower bound on the decay rate of $L^2$ solutions $u$ to $-\Delta u = Vu$ with $0\not\equiv V\in L^{\infty}(\mathbb{R}^d)$. We show that there exist a constant $M>0$ such that for any such $u$, for $R>0$ sufficiently large, one has \begin{equation*} \int_{B(0,R+1)\backslash \overline{B(0,R)}}|u(x)|^2 dx \geq M^{-1}R^{-4/3} \mathrm{e}^{-M \|V\|_{\infty}^{2/3} R^{4/3}}\|u\|^2_2. \end{equation*}

The semiclassical resolvent on conformally compact manifolds with variable curvature at infinity

ABSTRACTWe construct a semiclassical parametrix for the resolvent of the Laplacian acting on functions on nontrapping conformally compact manifolds with variable sectional curvature at infinity. We apply this parametrix to analyze the Schwartz kernel of the semiclassical resolvent and Poisson operator and to show that the semiclassical scattering matrix is a semiclassical Fourier Integral Operator of appropriate class that quantizes the scattering relation. We also obtain high energy estimates for the resolvent and show existence of resonance free strips of arbitrary height away from the imaginary axis. We then use the results of Datchev and Vasy on gluing semiclassical resolvent estimates to obtain semiclassical resolvent estimates on certain conformally compact manifolds with hyperbolic trapping.

Non-trapping estimates near normally hyperbolic trapping

In this paper we prove semiclassical resolvent estimates for operators with normally hyperbolic trapping which are lossless relative to non-trapping estimates but take place in weaker function spaces. In particular, we obtain non-trapping estimates in standard $L^2$ spaces for the resolvent sandwiched between operators which localize away from the trapped set $\Gamma$ in a rather weak sense, namely whose principal symbols vanish on $\Gamma$.

Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics

For a class of non-selfadjoint $h$--pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of the spectrum of the operator in an ${\cal O}(h)$--neighborhood of the origin. Moreover, when all the singular spaces are reduced to zero, we establish accurate semiclassical resolvent estimates of subelliptic type, which depend directly on algebraic properties of the Hamilton maps associated to the quadratic approximations of the principal symbol.

Non-Elliptic Quadratic Forms and Semiclassical Estimates for Non-selfadjoint Operators

We consider a class of pseudodifferential operators with a doubly characteristic point, where the quadratic part of the symbol fails to be elliptic but obeys an averaging assumption. Under suitable additional assumptions, semiclassical resolvent estimates are established, where the modulus of the spectral parameter is allowed to grow slightly more rapidly than the semiclassical parameter.

Resolvents and Seiberg-Witten representation for a Gaussian β-ensemble

The exact free energy of a matrix model always satisfies the Seiberg-Witten equations on a complex curve defined by singularities of the semiclassical resolvent. The role of the Seiberg-Witten differential is played by the exact one-point resolvent in this case. We show that these properties are preserved in the generalization of matrix models to β-ensembles. But because the integrability and Harer-Zagier topological recursion are still unavailable for β-ensembles, we must rely on the ordinary Alexandrov-Mironov-Morozov/Eynard-Orantin recursion to evaluate the first terms of the genus expansion. We restrict our consideration to the Gaussian model.

Propagation through trapped sets and semiclassical resolvent estimates

Motivé par l’étude des estimations de la résolvante dans la présence de capture, on démontre un théorème de propagation semiclassique dans un voisinage d’un sous-ensemble compact et invariant du flôt bicaractéristique, qui est isolé dans un sens convenable. Les exemples incluent un ensemble capté global et une trajectoire périodique isolée. Ceci est appliqué pour obtenir des estimations microlocales de la résolvante sans perte par rapport au cas non-captif.

Semiclassical resolvent estimates at trapped sets

We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs χ microlocally supported away from the trapping: ∥χR h (E+i0)χ∥=𝒪(h -1 ), a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, χ ˜, to be supported at the trapped set, giving ∥χR h (E+i0)χ ˜∥=𝒪(a(h)h -1 ) when the a priori bound is ∥χ ˜R h (E+i0)χ ˜∥=𝒪(a(h)h -1 ).