Related papers: Semiclassical resolvent bounds for compactly suppo…

We extend previous semi-classical resolvent estimates for real-valued compactly supported $L^\infty$ potentials to non-compactly supported $L^\infty$ potentials.

We prove a dispersive estimate for the evolution of Schroedinger operators H = -\Delta + V(x) in three dimensions. The potential should belong to the closure of bounded compactly-supported functions with respect to the golbal Kato norm.…

We consider the semiclassical Schr\"odinger operator $-h^2\partial_x^2+V(x)$ on a half-line, where $V$ is a compactly supported potential which is positive near the endpoint of its support. We prove that the eigenvalues and the purely…

We consider eigenvalue sums of Schr\"odinger operators $-\Delta+V$ on $L^2(\R^d)$ with complex radial potentials $V\in L^q(\R^d)$, $q<d$. We prove quantitative bounds on the distribution of the eigenvlaues in terms of the $L^q$ norm of $V$.…

It is well known that the resolvent of the free Schr\"odinger operator on weighted $L^2$ spaces has norm decaying like $\lambda^{-\frac{1}{2}}$ at energy $\lambda$. There are several works proving analogous high-frequency estimates for…

In this paper we study the decay estimates of the fourth order Schr\"{o}dinger operator $H=\Delta^{2}+V(x)$ on $\mathbb{R}^2$ with a bounded decaying potential $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ near the…

We consider Schr\"odinger operators $H=- \d^2/\d r^2+V$ on $L^2([0,\infty))$ with the Dirichlet boundary condition. The potential $V$ may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of $H$ is…

We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm…

In this paper we develop an abstract method to handle the problem of unique continuation for the Schr\"odinger equation $(i\partial_t+\Delta)u=V(x)u$. In general the problem is to find a class of potentials $V$ which allows the unique…

We consider the Dirichlet realization of the operator $-h^2\Delta+iV$ in the semi-classical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one dimensional setting, we obtain the complete asymptotic…

We prove a weighted Carleman estimate for a class of one-dimensional, self-adjoint Schr\"odinger operators $P(h)$ with low regularity electric and magnetic potentials, where $h > 0$ is a semiclassical parameter. The long range part of…

We prove limiting absorption resolvent bounds for the semiclassical Schr\"odinger operator with a repulsive potential in dimension $n\ge 3$, which may have a singularity at the origin. As an application, we obtain time decay for the…

We establish resolvent estimates that extend earlier results to a larger class of electric potentials $V\in L^\infty(\mathbb{R}^d;\mathbb{R})$, $d\ge 3$, and magnetic potentials $b\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$ such that $V(x),…

We prove a semiclassical resolvent estimate for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime. The proof relies on improved ellipticity properties for the symbol of the…

Consider operators $L_{V}:=\Delta + V$ in a bounded smooth domain $D$ in $R^N$. Assume that $V\in C^1(D)$ and $V$ may blow up at the boundary at most as $1/\delta^2$ where $\delta$ denotes distance to the boundary. Assume also that $L_{V}$…

We give an elementary proof of Burq's resolvent bounds for long range semiclassical Schroedinger operators. Globally, the resolvent norm grows exponentially in the inverse semiclassical parameter, and near infinity it grows linearly. We…

We consider the operator ${\mathcal A}_h=-\Delta+iV$ in the semi-classical $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the…

We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter $h$ tends to $0$. An example of such an operator is the shifted semiclassical Laplacian…

We prove dispersive bounds for fractional Schr\"odinger operators on $\mathbb R^n$ of the form $H=(-\Delta)^{\alpha}+V$ with $V$ a real-valued, decaying potential and $\alpha \notin\mathbb N$. We derive pointwise bounds on the resolvent…

We study the convergence of 1D Schr\"odinger ope\-rators $H_\varepsilon$ with the potentials which are regularizations of a class of pseudo-potentials having in particular the form $$ \alpha \delta'(x)+\beta…