arXiv subjects
Publications and source records attributed to Georgi Vodev.
We prove that for strictly concave domains much larger transmission eigenvalue-free regions exist. As a consequence, we get Weyl asymptotics for the total counting function with an almost optimal remainder term.
We obtain large regions free of eigenvalues of the interior transmission problem.
We prove a Weyl asymptotics $N(r) = c r^d + {\mathcal O}_ε(r^{d - κ+ ε})$, $\forall\, 0< ε\ll 1$, for the counting function $N(r) = \sharp\{λ_j \in {\mathbb C} \setminus \{0\}:\: |λ_j| \leq r^2\}$, $r>1$, of the interior transmission eigenvalues (ITE), $λ_j$. Here $0<κ\leq 1$ is such that there are no (ITE) in the region $\{λ\in {\mathbb C}:\: |{\rm Im}\:λ|\geq C(| {\rm Re}\:λ|+1)^{1-\fracκ{2}}\}$ for some $C>0$.
We prove optimal high-frequency resolvent estimates for perturbations by large magnetic and electric potentials
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.
We prove optimal high-frequency resolvent estimates for perturbations of the Laplacian by large long-range magnetic and electric potentials in all dimensions $n\ge 3$. As an application, we prove dispersive estimates for the corresponding wave group in the case $n=3$.
We prove dispersive estimates for the wave and Schrodinger groups associated to a second-order elliptic self-adjoint operator depending on a semi-classical parameter. Applications are made to non-trapping metric perturbations and to perturbations by a magnetic potential.
We prove optimal dispersive estimates for the wave group $e^{it\sqrt{-Δ+V}}$ for a class of real-valued potentials $V\in C^{(n-3)/2}({\bf R}^n)$, $4\le n\le 7$, such that $\partial_x^αV(x)=O(|x|^{-δ})$ for $|x|>1$, where $δ>(n+1)/2$.
We prove optimal dispersive estimates at high frequency for the Schrodinger group with real-valued potentials $V(x)=O(|x|^{-δ})$, $δ>n-1$, and $V\in C^k({\bf R}^n$, $k>k_n$, where $n\ge 4$ and $(n-3)/2\le k_n<n/2$. We also give a sufficient condition in terms of $L^1\to L^\infty$ bounds for the formal iterations of Duhamel's formula, which might be satisfied for potentials of less regularity.
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 free of eigenvalues regions. As an application, we get exponential decay of the energy of the solutions of the correpsonding mixed boundary value problems.
We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrodinger group $e^{it(-Δ+V)}$ for a class of real-valued potentials $V\in C^k(R^n)$ with $k>(n-3)/2$, where $n=4,5$.
We prove dispersive estimates for the low frequency part of the Schrodinger group for a large class of potentials in dimensions greater or equal to four. As a consequence, we extend the result of Journe, Sofer and Sogge to a larger class of potentials. In this revised version a mistake in the proof of the estimate (B.4) is removed.
We obtain free of resonances regions for the elasticity system in the exterior of a strictly convex body with dissipative boundary conditions under some natural assumptions on the behaviour of the geodesics on the boundary.
We prove dispersive estimates for solutions to the Schrodinger equation with a real-valued potential $V\in L^\infty({\bf R}^n)$, $n\ge 4$, satisfying $V(x)=O(|x|^{-(n+2)/2-ε})$, $|x|>1$, $ε>0$.
We prove dispersive estimates for solutions to the wave equation with a real-valued potential $V\in L^\infty({\bf R}^n)$, $n\ge 4$, satisfying $V(x)=O(|x|^{-(n+1)/2-ε})$, $|x|>1$, $ε>0$.
We obtain large time decay estimates on weighted $L^p$ spaces for solutions to the wave equation with real-valued potential $V(x)=O(|x|^{-2-a})$, $a>0$, for $|x|>1$.