目前的研究集中于数学物理方面，特别是离散薛定谔算子的谱理论。近年的工作致力于使用动力系统和分形几何的工具研究算子的谱的维数。对具Sturm势的算子：证明在大势强度情形，算子的谱以及状态密度测度的Hausdorff维数关于几乎所有的频率取常值，且状态密度测度有精确维数性质。对临界情形的几乎Mathieu算子：首次找到一个频率集，使得对取值于其中的频率，对应算子的谱具有正Hausdorff维数。对具倍周期势的算子：对任意势强度得到谱的一个编码映射，借助这个编码得到谱的维数的一个绝对正下界。对具Thue-Morse势的算子：得到该算子的谱的维数的一个绝对下界；证明了该算子存在无界迹轨道，并进而证明该算子呈现伪局部化现象。
[1] Liu Qinghui; Qu Yanhui; Yao Xiao, The spectrum of period-doubling Hamiltonian. Comm. Math. Phys. 394 (2022), no. 3, 1039–1100.
[2] Helffer Bernard; Liu Qinghui; Qu Yanhui; Zhou Qi, Positive Hausdorff dimensional spectrum for the critical almost Mathieu operator. Comm. Math. Phys. 368 (2019), no. 1, 369–382.
[3] Qu Yan-Hui, Exact-dimensional property of density of states measure of Sturm Hamiltonian, Int. Math. Res. Not.( 2018), no. 17, 5417–5454.
[4] Liu Qinghui; Qu Yanhui; Yao Xiao, Unbounded Trace Orbits of Thue–Morse Hamiltonian. J. Stat. Phys. 166 (2017), no. 6, 1509–1557.
[5] Qu Yan-Hui, The spectral properties of the strongly coupled Sturm Hamiltonian of eventually constant type. Ann. Henri Poincaré 17 (2016), no. 9, 2475–2511.
[6] Liu Qinghui; Qu Yanhui, On the Hausdorff dimension of the spectrum of the Thue-Morse Hamiltonian. Comm. Math. Phys. 338 (2015), no. 2, 867–891.
[7] Liu Qing-Hui; Qu Yan-Hui; Wen Zhi-Ying, The fractal dimensions of the spectrum of Sturm Hamiltonian. Adv. Math. 257 (2014), 285–336.
