Decay estimates for higher order elliptic operators

2019/04/28 by Hongliang Feng, Avy Soffer, Feng, Hongliang +5
Mathematics · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1904.12275

Abstract

This paper is mainly devoted to study time decay estimates of the higher-order Schrödinger type operator H=(-Δ)^m+V(x) in Rⁿ for n>2m and m\inN. For certain decay potentials V(x), we first derive the asymptotic expansions of resolvent R_V(z) near zero threshold with the presence of zero resonance or zero eigenvalue, as well identify the resonance space for each kind of zero resonance which displays different effects on time decay rate. Then we establish Kato-Jensen type estimates and local decay estimates for higher order Schrödinger propagator e^-itH in the presence of zero resonance or zero eigenvalue. As a consequence, the endpoint Strichartz estimate and L^p-decay estimates can also be obtained. Finally, by a virial argument, a criterion on the absence of positive embedded eigenvalues is given for (-Δ)^m+V(x) with a repulsive potential.

Citations

• Spectral properties of Schrödinger operators and time-decay of the wave functions
• Time decay for solutions of Schr�dinger equations with rough and time-dependent potentials

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