Georgi Vodev

Local energy decay of solutions to the wave equation for nontrapping metrics

We prove uniform local energy decay estimates of solutions to the wave equation on unbounded Riemannian manifolds with nontrapping metrics. These estimates are derived from the properties of the resolvent at high frequency. Applications to a class of asymptotically Euclidean manifolds as well as to perturbations by non-negative long-range potentials are given.

Dispersive estimates for the Schrodinger equation in dimensions four and five

We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrodinger group e it(−�+V ) for a class of real-valued potentials V ∈ C k (R n ), V (x) = O(h xi −δ ), where n = 4,5, k > (n − 3)/2, δ > 3 if n = 4 and δ > 5 if n = 5.

Dispersive Estimates of Solutions to the Wave Equation with a Potential in Dimensions n ≥ 4

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L ∞(R n ), n ≥ 4, satisfying V(x) = O(⟨x⟩−(n+1)/2−ε), ε > 0.

ASYMPTOTICS OF THE NUMBER OF RESONANCES IN THE TRANSMISSION PROBLEM

We consider the resonances for the transmission problem associated with a strictly convex transparent obstacle. Under some natural assumptions we show that there is a free of resonances region in the complex upper half plane given by {C ≤ Im λ ≤ C 1|λ|1/3 − C 2}, where C, C 1 and C 2 are positive constants. Moreover, we obtain asymptotics for the number of resonances counted with multiplicities in the region {0 < Im λ ≤ C, 0 < Re λ ≤ r} as r → ∞, where C > 0 is the same constant as above. *Partially supported by CNPq (Brazil).

Sharp Bounds on the Number of Resonances for Conformally Compact Manifolds with Constant Negative Curvature Near Infinity

Abstract The purpose of this article is to prove a sharp bound on the number of resonances for the Laplacian on conformally compact manifolds with constant negative curvature near infinity, thus improving the polynomial bound of Guillopé and Zworki (Guillopé, L., Zworski, M. ([1995b]). Polynomial bound on the number of resonances for some complete spaces of constant negative curvature near infinity. Asympt. Anal. 11:1–22).

Smoothing Effect for the Regularized Schrödinger Equation with Non-Controlled Orbits

We prove that the geometric control condition is not necessary to obtain the smoothing effect and the uniform stabilization for the strongly dissipative Schrödinger equation.

High Frequency Resolvent Estimates and Energy Decay of Solutions to the Wave Equation

Let (M, g) be an n-dimensional non-compact, connected Riemannian manifold with a Riemannian metric g of class C(M) and a compact C-smooth boundary ∂M (which may be empty), of the form M = X0 ∪X, where X0 is a compact, connected Riemannian manifold with a metric g|X0 of class C (X0) with a compact boundary ∂X0 = ∂M ∪ ∂X, ∂M ∩ ∂X = ∅, X = [r0,+∞)× S, r0 1, with metric g|X := dr + σ(r). Here (S, σ(r)) is an n − 1 dimensional compact Riemannian manifold without boundary equipped with a family of Riemannian metrics σ(r) depending smoothly on r which can be written in any local coordinates θ ∈ S in the form

Optimal Dispersive Estimates for the Wave Equation with Potentials in Dimensions 4 ≤ n ≤ 7

We prove optimal dispersive estimates for the wave group for a class of real-valued potentials , 4 ≤ n ≤ 7, such that , , .

Boundary stabilization of transmission problems

We study the transmission problem in bounded domains with dissipative boundary conditions. Under some natural assumptions, we prove uniform bounds of the corresponding resolvents on the real axis at high frequency and, as a consequence, we obtain regions free of eigenvalue. To this end, we extend the result of Cardoso et al. [“Distribution of resonances and local energy decay in the transmission problem. II,” Math. Res. Lett. 6, 377 (1999)] under more general assumptions. As an application, we get exponential decay of the energy of the solutions of the corresponding mixed boundary value problems.

Resolvent estimates for perturbations by large magnetic potentials

We prove optimal high-frequency resolvent estimates for self-adjoint operators of the form G = −Δ + ib(x) · ∇ + i∇ · b(x) + V(x) on L2(Rn), n ⩾ 3, where b(x) and V(x) are large magnetic and electric potentials, respectively. No continuity of the magnetic potential is assumed.