On 4-order Schrödinger operator and Beam operator

Abstract

We show that there is no localization for the 4-order Schrödinger operator {{\displaystyle \mathcal{S}}}_{t,\mathrm{4}}f and Beam operator {{\displaystyle B}}_{t}f, more precisely, on the one hand, we show that the 4-order Schrödinger operator {{\displaystyle S}}_{t,\mathrm{4}}f does not converge pointwise to zero as t\to \mathrm{0} provided f\in H^s(ℝ) with compact support and 0<s<1/4 by constructing a counterexample in ℝ. On the other hand, we show that the Beam operator Btf also has the same property with the 4-order Schrödinger operator {{\displaystyle \mathcal{S}}}_{t,\mathrm{4}}f. Hence, we find that the Hausdorff dimension of the divergence set for {{\displaystyle \mathcal{S}}}_{t,\mathrm{4}}f and {{\displaystyle B}}_{t}f is {{\displaystyle \alpha }}_{\mathrm{1},{{\displaystyle S}}_{\mathrm{4}}}(s)={{\displaystyle \alpha }}_{\mathrm{1},B}(s)=\mathrm{1} as 0<s<1/4.

Key words： 4-Order Schrödinger operator; Beam operator; localization; Sobolev space

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