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Register a new userA Limiting absorption principle for high-order Schrodinger operators in critical spaces
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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
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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.
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.
In this paper we prove some multi-linear Strichartz estimates for solutions to the linear Schrodinger equations on torus $T^n$. Then we apply it to get some local well-posed results for nonlinear Schrodinger equation in critical $H^{s}(T^n)$ spaces. As by-products, the energy critical global well-posed results and energy subcritical global well-posed results with small initial data are also obtained.
We establish that trace inequalities $$|D^{k-1}u|_{L^{frac{n-s}{n-1}}(mathbb{R}^{n},dmu)} leq c |mu|_{L^{1,n-s}(mathbb{R}^{n})}^{frac{n-1}{n-s}}|mathbb{A}[D]u|_{L^{1}(mathbb{R}^{n},dmathscr{L}^{n})}$$ hold for vector fields $uin C^{infty}(mathbb{R}^{n};mathbb{R}^{N})$ if and only if the $k$-th order homogeneous linear differential operator $mathbb{A}[D]$ on $mathbb{R}^{n}$ is elliptic and cancelling, provided that $s<1$, and give partial results for $s=1$, where stronger conditions on $mathbb{A}[D]$ are necessary. Here, $|mu|_{L^{1,lambda}}$ denotes the $(1,lambda)$-Morrey norm of the measure $mu$, so that such traces can be taken, for example, with respect to the Hausdorff measure $mathscr{H}^{n-s}$ restricted to fractals of codimension $0<s<1$. The above class of inequalities give a systematic generalisation of Adams trace inequalities to the limit case $p=1$ and can be used to prove trace embeddings for functions of bounded $mathbb{A}$-variation, thereby comprising Sobolev functions and functions of bounded variation or deformation. We moreover establish a multiplicative version of the above inequality, which implies ($mathbb{A}$-)strict continuity of the associated trace operators on $text{BV}^{mathbb{A}}$.
We study the wellposedness of Cauchy problem for the fourth order nonlinear Schrodinger equations ipartial_t u=-epsDelta u+Delta^2 u+P((partial_x^alpha u)_{abs{alpha}ls 2}, (partial_x^alpha bar{u})_{abs{alpha}ls 2}),quad tin Real, xinReal^n, where $epsin{-1,0,1}$, $ngs 2$ denotes the spatial dimension and $P(cdot)$ is a polynomial excluding constant and linear terms.
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