On the 𝐿^{𝑝} boundedness of the wave operators for fourth order Schrödinger operators,Transactions of the American Mathematical Society

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On the 𝐿^{𝑝} boundedness of the wave operators for fourth order Schrödinger operators
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2021-03-24 , DOI: 10.1090/tran/8377
Michael Goldberg , William R. Green

Abstract:We consider the fourth order Schrödinger operator $H=\Delta ^2+V(x)$ in three dimensions with real-valued potential

. Let $H_0=\Delta ^2$ , if

decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous spectrum of

then the wave operators $W_{\pm }= s$ $\text {\,--}\lim _{t\to \pm \infty } e^{itH}e^{-itH_0}$ extend to bounded operators on $L^p(\mathbb{R}^3)$ for all $H = \ Delta ^ 2 + V（x）$

$H_0 = \ Delta ^ 2$

$W _ {\ pm} = s$ $\ text {\，-} \ lim _ {t \ to \ pm \ infty} e ^ {itH} e ^ {-itH_0}$ $L ^ p（\ mathbb {R} ^ 3）$ $\ mathbb {K} [x_0，\ ldots，x_n]$ $\ mathbb {K}$ $\ mathrm {Supp}（\ Sigma）$ $\西格玛$ $\ mathrm {Supp}（\ Sigma）$

$1 \ leq a \ leq \ vert \ Sigma \ vert$