Let $M^{\circ }$ be a complete noncompact manifold of dimension at least 3 and $g$ an asymptotically conic metric on $M^{\circ }$, in the sense that $M^{\circ }$ compactifies to a manifold with boundary $M$ so that $g$ becomes a scattering metric on $M$. We study the resolvent kernel ${(P+k^2)}^{-1}$ and Riesz transform $T$ of the operator $P={\Delta }_g+V$, where ${\Delta }_g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function smooth on $M$ and vanishing at the boundary.In our first paper we assumed that $P$ has neither zero modes nor a zero-resonance and showed...

We consider the 3D Schrödinger operator $H=H_0+V$ where $H_0={(-i\nabla -A)}^2-b$, $A$ is a magnetic potential generating a constant magneticfield of strength $b\>0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a meromorphic extension from the upper half plane to an appropriate Riemann surface $ℳ$, and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed...

Let E,F be Banach spaces where F = E’ or vice versa. If F has the approximation property, then the space of nuclearly entire functions of bounded type, ${\mathscr{H}}_{Nb}\left(E\right)$, and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on ${\mathscr{H}}_{Nb}\left(E\right)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $\phi \left(D\right){\equiv }^t\phi$, φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on ${\mathscr{H}}_{Nb}\left(E\right)$ is PDE-preserving for a set ℙ ⊆ Exp(F) if it...

The scattering matrix is defined on a perturbed stratified medium. For a class of perturbations, its main part at fixed energy is a Fourier integral operator on the sphere at infinity. Proving this is facilitated by developing a refined limiting absorption principle. The symbol of the scattering matrix determines the asymptotics of a large class of perturbations.

Strong asymptotic completeness is shown for a pair of Schrödinger type operators on a cylindrical Lipschitz domain. A key ingredient is a limiting absorption principle valid in a scale of weighted (local) Sobolev spaces with respect to the uniform topology. The results are based on a refined version of Mourre’s method within the context of pseudo-selfadjoint operators.

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show that the...

Let $A_{𝒩}$ be the symmetric operator given by the restriction of $A$ to $𝒩$, where $A$ is a self-adjoint operator on the Hilbert space $\mathscr{H}$ and $𝒩$ is a linear dense set which is closed with respect to the graph norm on $D\left(A\right)$, the operator domain of $A$. We show that any self-adjoint extension $A_{\Theta }$ of $A_{𝒩}$ such that $D\left(A_{\Theta }\right)\cap D\left(A\right)=𝒩$ can be additively decomposed by the sum $A_{\Theta }=\overline{A}+T_{\Theta }$, where both the operators $\overline{A}$ and $T_{\Theta }$ take values in the strong dual of $D\left(A\right)$. The operator $\overline{A}$ is the closed extension of $A$ to the whole $\mathscr{H}$ whereas $T_{\Theta }$ is explicitly written in terms...