Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks

Abstract

We prove the time decay estimates $L^1(\mathcal{R})\to L^{\infty }(\mathcal{R}),$ where $\mathcal{R}$ is an infinite star-shaped network, for the Schrödinger group $e^{it(-\frac{d^2}{d{x}^2}+V)}$ for real-valued potentials $V$ satisfying some regularity and decay assumptions. Further we show that the solution for initial conditions with a lower cutoff frequency tends to the free solution, if the cutoff frequency tends to infinity.

This paper is an improved and extended version of "A dispersive estimate for the Schrödinger operator in star-shaped networks"

References in corpus (2)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• Dispersion for 1-d Scrodinger and wave equation with BV coefficients

Cited by in corpus (3)

• Explicit error estimates for the stationary phase method I: The influence of amplitude singularities
• Explicit error estimates for the stationary phase method II: Interaction of amplitude singularities with stationary points
• Asymptotic estimates of oscillatory integrals with general phase and singular amplitude: Applications to dispersive equations