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Register a new userLimiting absorption principle for some long range perturbations of Dirac systems at threshold energies
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We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to study a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non self-adjoint operators, which we develop in appendix. We also discuss some applications to the dispersive Helmholzt model in the quantum regime.
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We consider the Landau Hamiltonian perturbed by a long-range electric potential $V$. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we obtain an estimate of the rate of the shrinking of these clusters to the Landau levels as the number of the cluster $q$ tends to infinity. Further, we assume that there exists an appropriate $V$, homogeneous of order $-rho$ with $rho in (0,1)$, such that $V(x) = V(x) + O(|x|^{-rho - epsilon})$, $epsilon > 0$, as $|x| to infty$, and investigate the asymptotic distribution of the eigenvalues within a given cluster, as $q to infty$. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the mean-value transform of $V$.
We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on $mathbb{R}^3$, which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of $-frac{1}{2}$-Killing spinors on the round sphere $mathbb{S}^3$.
In this paper, we prove a limiting absorption principle for high-order Schrodinger operators with a large class of potentials which generalize some results by A. Ionescu and W. Schlag. Our main idea is to handle the boundary operators by the restriction theorem of Fourier transform. Two key tools we use in this paper are the Stein--Tomas theorem in Lorentz spaces and a sharp trace lemma given by S. Agmon and L. Hormander
We use a Lagrangian perspective to show the limiting absorption principle on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. More precisely we show that, for non-zero spectral parameter, the `on spectrum, as well as the `off-spectrum, spectral family is Fredholm in function spaces which encode the Lagrangian regularity of generalizations of `outgoing spherical waves of scattering theory, and indeed this persists in the `physical half plane.
Let $H$ denote the harmonic oscillator Hamiltonian on $mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schrodinger propagator $U(t)=e^{-itH},$ and find that while $operatorname{singsupp} operatorname{Tr} U(t) subset 2 pi mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d geq 2$ for which the singularities of $operatorname{Tr} U(t)$ at nonzero multiples of $2 pi$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(lambda^{d-1})$, improving in these cases the $O(lambda^{d-1})$ remainder previously established by Helffer--Robert.
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