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Register a new userDispersion Estimates for Spherical Schrodinger Equations with Critical Angular Momentum
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We derive a dispersion estimate for one-dimensional perturbed radial Schrodinger operators where the angular momentum takes the critical value $l=-frac{1}{2}$. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.
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We investigate the dependence of the $L^1to L^infty$ dispersive estimates for one-dimensional radial Schro-din-ger operators on boundary conditions at $0$. In contrast to the case of additive perturbations, we show that the change of a boundary condition at zero results in the change of the dispersive decay estimates if the angular momentum is positive, $lin (0,1/2)$. However, for nonpositive angular momenta, $lin (-1/2,0]$, the standard $O(|t|^{-1/2})$ decay remains true for all self-adjoint realizations.
We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Dirac equation. To this end we develop basic scattering theory and establish a limiting absorption principle for discrete perturbed Dirac operators.
We show that for a one-dimensional Schrodinger operator with a potential whose (j+1)th moment is integrable the jth derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrodinger equation in the resonant case.
Let $Sigmasubsetmathbb{R}^d$ be a $C^infty$-smooth closed compact hypersurface, which splits the Euclidean space $mathbb{R}^d$ into two domains $Omega_pm$. In this note self-adjoint Schrodinger operators with $delta$ and $delta$-interactions supported on $Sigma$ are studied. For large enough $minmathbb{N}$ the difference of $m$th powers of resolvents of such a Schrodinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(mathbb{R}^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(Sigma)$.
We show that for a one-dimensional Schrodinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive dispersion estimates for solutions of the associated Schrodinger and Klein-Gordon equations. In particular, we remove the additional decay conditions in the case where a resonance is present at the edge of the continuous spectrum.
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