L^p boundedness of wave operators for higher order schrödinger operators with threshold eigenvalues

2025/06/19 by M. Burak Erdogan, William R. Green, Erdogan, M. Burak +3 · 1 citation
#math.AP #math.SP

paper · pdf · doi:10.48550/arxiv.2506.16378

Abstract

We consider the higher order Schrödinger operator H=(-Δ)^m+V(x) in n dimensions with real-valued potential V when n>2m, m∈ \mathbb N when H has a threshold eigenvalue. We adapt our recent results for m≥ 1 when n>4m to lower dimensions 2m<n≤ 4m to show that when H has a threshold eigenvalue and no resonances, the wave operators are bounded on L^p(\mathbb Rⁿ) for the natural range 1≤ p<(2n)/(n-1) when n is odd and 1≤ p<(2n)/(n-2) when n is even. We further show that if the zero energy eigenfunctions are orthogonal to x^αV(x) for all |α|<k₀, then the wave operators are bounded on 1≤ p<(n)/(2m-k₀) when k₀<2m in all dimensions n>2m. The range is p∈ [1,∞) and p∈[1,∞] when k₀=2m and k₀>2m respectively. The proofs apply in the classical m=1 case as well and streamlines existing arguments in the eigenvalue only case, in particular the L^∞(\mathbb Rⁿ) boundedness is new when n>3.

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