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Decay estimates for fourth-order Schrödinger operators in dimension two. (English) Zbl 1511.35137

Summary: In this paper we study the decay estimates of the fourth order Schrödinger operator \(H = \Delta^2 + V(x)\) on \(\mathbb{R}^2\) with a bounded decaying potential \(V(x)\). We first deduce the asymptotic expansions of resolvent of \(H\) near zero threshold in the presence of resonances or eigenvalue, and then use them to establish the \(L^1 - L^\infty\) decay estimates of \(e^{- i t H}\) generated by the fourth order Schrödinger operator \(H\). Our methods used in the decay estimates depend on Littlewood-Paley decomposition and oscillatory integral theory. Moreover, we also classify these zero resonances as the distributional solutions of \(H \phi = 0\) in suitable weighted spaces. Due to the degeneracy of \(\operatorname{\Delta}^2\) at zero threshold, we remark that the asymptotic expansions of resolvent \(R_V( \lambda^4)\) and the classifications of resonances are much more involved than Schrödinger operator \(- \Delta + V\) in dimension two.

MSC:

35J30 Higher-order elliptic equations
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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