Related papers: Improved resolvent bounds for radial potentials. I…

We consider a non-self-adjoint $h$-pseudodifferential operator $P$ in the semi-classical limit ($h\to 0$). If $p$ is the leading symbol, then under suitable assumptions about the behaviour of $p$ at infinity, we know that the resolvent…

We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrodinger group $e^{it(-\Delta+V)}$ for a class of real-valued potentials $V\in C^k(R^n)$ with $k>(n-3)/2$, where $n=4,5$.

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that the wave operators are bounded on…

In this paper we consider the vector-valued Schr\"{o}dinger operator $-\Delta + V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is…

We consider the Schr\"odinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold $0$.…

We prove $L^p\to L^{p'}$ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension $n$ in the endpoint case $p=2(n+1)/(n+3)$. It has the same behavior with respect to the spectral parameter $z$…

We prove optimal high-frequency resolvent estimates for perturbations by large magnetic and electric potentials

We study the eigenvalues of Schr\"odinger operators with complex potentials in odd space dimensions. We obtain bounds on the total number of eigenvalues in the case where $V$ decays exponentially at infinity.

We prove dispersive estimates for the linear Schr\"odinger evolution associated to an operator -\Delta + V, where the potential is a signed measure of fractal dimension at least 3/2.

We study the $L^\infty(\mathbb{R}^d)$ boundedness for Riesz transforms of the form ${V^{a}(-\frac12\Delta+V)^{-a}},$ where $a > 0$ and $V$ is a non-negative potential with power growth acting independently on each coordinate. We factorize…

In this paper we prove a subelliptic resolvent estimate for a class of semiclassical non-self-adjoint Schr\"odinger operators with purely imaginary potentials when the spectral parameter is in a parabolic neighborhood of the imaginary axis.

We study spectral approximations of Schr\"odinger operators $T=-\Delta+Q$ with complex potentials on $\Omega=\mathbb{R}^d$, or exterior domains $\Omega\subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of…

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d>2 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. We show for generic potentials V that the number of…

In this paper, let $L=L_{0}+V$ be a Schr\"{o}dinger type operator where $L_{0}$ is higher order elliptic operator with complex coefficients in divergence form and $V$ is signed measurable function, under the strongly subcritical assumption…

The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially we prove a number of lower and upper bounds for the resolvent of such operators. We consider two…

This article addresses the microlocalization of eigenfunctions for the semiclassical Schr\"odinger operator $-h^2\Delta+V$ on closed Riemann surfaces with real bounded potentials. Our primary aim is to establish quantitative bounds on the…

We discuss the eigenvalues $E_j$ of Schr\"odinger operators $-\Delta+V$ in $L^2(\mathbb R^d)$ with complex potentials $V\in L^p$, $p<\infty$. We show that (A) $\mathrm{Re} E_j\to\infty$ implies $\mathrm{Im} E_j\to 0$, and (B) $\mathrm{Re}…

We prove Strichartz estimates for the Schroedinger operator $H = -\Delta + V(t,x)$ with time-periodic complex potentials $V$ belonging to the scaling-critical space $L^{n/2}_x L^\infty_t$ in dimensions $n \ge 3$. This is done directly from…

In this note, we prove an optimal universal lower bound on the truncated resolvent for semiclassical Schroedinger operators near a trapping energy. In particular, this shows that known upper bounds for hyperbolic trapping are optimal. The…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0,$ we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity.…