Rafe Mazzeo, Andras Vasy
Let $(M,g)$ be a globally symmetric space of noncompact type, of arbitrary rank, and $Δ$ its Laplacian. We prove the existence of a meromorphic continuation of the resolvent $(Δ-\ev)^{-1}$ across the continuous spectrum to a Riemann surface multiply covering the plane. The methods are purely analytic and are adapted from quantum $N$-body scattering.
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We consider optimization problems for cost functionals which depend on the negative spectrum of Schrödinger operators of the form $-Δ+V(x)$, where $V$ is a potential, with prescribed compact support, which has to be determined. Under suitable assumptions the existence of an optimal potential is shown. This can be applied to interesting cases such as costs functions involving finitely many negative eigenvalues.
We show the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recently, roughly by extending the problem across the boundary of the compactification of the asymptotically hyperbolic space in a suitable manner. The main novelty is that the non-scalar nature of the operator is dealt with by relating it to a problem on an asymptotically Minkowski spa…
In this paper we construct a parametrix for the high-energy asymptotics of the analytic continuation of the resolvent on a Riemannian manifold which is a small perturbation of the Poincaré metric on hyperbolic space. As a result, we obtain non-trapping high energy estimates for this analytic continuation.
In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. In our previous work we described the resolvent, and specifically the asymptotic behavior of the Green's function, on SL(3)/SO(3) using methods from three-particle scattering. Here we extend the technique of complex scaling to symmetric spaces to show that the resolvent continues analytically across the spectrum.
Let $s_0 < 0$ be the abscissa of absolute convergence of the dynamical zeta function $Z(s)$ for several disjoint strictly convex compact obstacles $K_i \subset \R^N, i = 1,..., κ_0,\: \ka_0 \geq 3,$ and let $R_χ(z) = χ(-Δ_D - z^2)^{-1}χ,\: χ\in C_0^{\infty}(\R^N),$ be the cut-off resolvent of the Dirichlet Laplacian $-Δ_D$ in $Ω= \bar{\R^N \setminus \cup_{i = 1}^{k_0} K_i}$. We prove that there exists $σ_1 < s_0$ such that $Z(s)$ is analytic for $\Re (s) \geq σ_1$ and the cut-off resolvent $R_χ(z)$ has an analytic continuation for $\Im (z) < - σ_1,\: |\Re (z)| \geq C > 0.$
We extend Vasy's results on semiclassical high energy estimates for the meromorphic continuation of the resolvent for asymptotically hyperbolic manifolds to metrics that are not necessarily even. Vasy's method gives the meromorphic continuation of the resolvent and high energy estimates in strips, assuming that the geodesic flow is non-trapping, without having to construct a parametrix. We prove that the size of the strip to which the resolvent can be extended meromorphically is the same obtained by Guillarmou.
In this paper we develop an abstract method to handle the problem of unique continuation for the Schrödinger equation $(i\partial_t+Δ)u=V(x)u$. In general the problem is to find a class of potentials $V$ which allows the unique continuation. The key point of our work is to make a direct link between the problem and the weighted $L^2$ resolvent estimates $\|(-Δ-z)^{-1}f\|_{L^2(|V|)}\leq C\|f\|_{L^2(|V|^{-1})}$. We carry out it in an abstract way, and thereby we do not need to deal with each of the potential classes. To do so, we will make use of limiting absorption principle and Kato $H$-smooth…
Consider the wave equation $\partial_t^2u-Δ_xu+V(t,x)u=0$, where $x\in\R^n$ with $n\geq3$ and $V(t,x)$ is $T$-periodic in time and decays exponentially in space. Let $ U(t,0)$ be the associated propagator and let $R(θ)=e^{-D<x>}(U(T,0)-e^{-iθ})^{-1}e^{-D<x>}$ be the resolvent of the Floquet operator $U(T,0)$ defined for $\im(θ)>BT $ with $B>0$ sufficiently large. We establish a meromorphic continuation of $R(θ)$ from which we deduce the asymptotic expansion of $e^{-(D+ε)<x>}U(t,0)e^{-D<x>}f$, where $f\in \dot{H}^1(\R^n)\times L^2(\R^n)$, as $t\to+\infty$ with a remainder term whose energy deca…
Pollicott-Ruelle resolvent and Sobolev regularity
本文研究Anosov流的Pollicott–Ruelle预解式，在一般向量丛提升情形下确定其在各向异性Sobolev空间中亚纯延拓的正则性阈值，并联系关联衰减所需正则性。
On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric tensors, a similar result is proven for (convex cocompact) quotients of hyperbolic space.
In this paper we describe a new method for analyzing the Laplacian on asymptotically hyperbolic spaces, which was introduced recently by the author. This new method in particular constructs the analytic continuation of the resolvent for even metrics (in the sense of Guillarmou), and gives high energy estimates in strips. The key idea is an extension across the boundary for a problem obtained from the Laplacian shifted by the spectral parameter. The extended problem is non-elliptic -- indeed, on the other side it is related to the Klein-Gordon equation on an asymptotically de Sitter space -- bu…
A Measure-Valued Obstacle Problem for an Obliquely Reflected Diffusion with a Max-Type Payoff
针对非负象限中具有非光滑最大值型收益的斜反射最优停止问题，提出测度值势论形式的障碍问题公式，推导反射Itô-Tanaka恒等式，并证明验证定理。
We develop the complex scaling for a manifold with an asymptotically cylindrical end under an assumption on the analyticity of the metric with respect to the axial coordinate of the end. We allow for arbitrarily slow convergence of the metric to its limit at infinity, and prove a variant of the Aguilar-Balslev-Combes theorem for the Laplacian $Δ$ on functions. In the case of a manifold with (noncompact) boundary it is either the Dirichlet or the Neumann Laplacian. We introduce resonances as the discrete non-real eigenvalues of non-selfadjoint operators, obtained as deformations of the Laplacia…
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