In two dimensions, for scaling-critical electromagnetic Schrödinger operators with mean flux outside 1/2 Z (or with symmetric electric potential in the resonant case), the newly defined intertwining operators are L^p-bounded and transfer key estimates from the magnetic-only operator.
$L^p$-continuity of wave operators for higher order Schr\"odinger operators with threshold eigenvalues in high dimensions
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abstract
We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that when $H$ has a threshold eigenvalue the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{n}{2m}$ in both even and odd dimensions. The approach used works without distinguishing even and odd cases, and matches the range of boundedness in the classical case when $m=1$. The proof applies in the classical $m=1$ case as well and simplifies the argument.
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Intertwining operators beyond the Stark Effect
In two dimensions, for scaling-critical electromagnetic Schrödinger operators with mean flux outside 1/2 Z (or with symmetric electric potential in the resonant case), the newly defined intertwining operators are L^p-bounded and transfer key estimates from the magnetic-only operator.