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 [9] 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....

We discuss the relationships between the notion of intrinsic ultracontractivity, parabolic Harnack principle, compactness of the 1-resolvent of the Neumann Laplacian, and non-trap property for Euclidean domains with finite Lebesgue measure. In particular, we give an answer to an open problem raised by Davies and Simon in 1984 about the possible relationship between intrinsic ultracontractivity ...

For a class of manifolds X that includes quotients of real hyperbolic (n + 1)-dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for X. In order to carry out the proof, we use Shmuel Agmon’s ...

Keywords: Drazin inverse Anti-triangular operator matrix Resolvent expansion a b s t r a c t This paper deals with the anti-triangular operator matrix M ¼ A B C 0 with A 2 ¼ A and CA p B ¼ 0. Using the resolvent expansion technique, we obtain the explicit representation of the Drazin inverse of M, in terms of its entries and the Drazin inverses of the entries and their compositions. The result ...

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 [9] 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....

In this paper we prove semiclassical resolvent estimates for operators with normally hyperbolic trapping which are lossless relative to nontrapping estimates but take place in weaker function spaces. In particular, we obtain non-trapping estimates in standard L2 spaces for the resolvent sandwiched between operators which localize away from the trapped set Γ in a rather weak sense, namely whose ...

We present results concerning resolvent estimates for the linear operator associated with the system of differential equations governing perturbations of the Couette flow. We prove estimates on the norm of the resolvent of this operator showing this norm to be proportional to the Reynolds number R for a region of the complex plane. Numerical approximations indicate that the same dependence is v...

In this paper, we use the resolvent operator to suggest and analyze two new numerical methods for solving general mixed quasi variational inequalities coupled with new directions and new step sizes. Under certain conditions, the global convergence of the both methods is proved. Our results can be viewed as significant extensions of the previously known results for general mixed quasi variationa...

We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be...

The linearized Vlasov equation for a plasma system in an external constant magnetic field and the corresponding linear Vlasov operator are studied. The solution of the Vlasov equation is found by the resolvent method. The spectrum and eigenfunctions of the Vlasov operator are also found. The spectrum of this operator consists of two parts: one is continuous and real; the other is discrete and c...