This paper is concerned with the nonperiodic Hamiltonian system $u^{\prime }=JL\left(t\right)u+J{H}_u^{\prime }(t,u),t\in \mathbb{R}.$Inspired by Dolbeault, Esteban and Séré [On the eigenvalues of operators with gaps. Application to Dirac operators. J Funct Anal. 2000;174:208–226], we will reduce the associated linear system to a new linear system with finite Morse index. We will develop an index theory and define the relative Morse index by investigating the Morse index of the reduced linear Hamiltonian system. Combining the index theory with a generalized linking theorem developed by Bartsch and Ding [Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math Nachr. 2006;279:1267–1288.], the existence and multiplicity of solutions are obtained for asymptotically quadratic nonlinearity.

这篇论文关注的是非周期哈密顿系统${你}^{‘}=杰\mathrm{大号}\left(吨\right)你+杰H_{你}^{‘}(吨,你),吨\in \mathbb{R}.$受 Dolbeault、Esteban 和 Séré 的启发 [关于有间隙算子的特征值。狄拉克算子的应用。J 功能肛门。2000;174:208–226]，我们将相关的线性系统简化为具有有限莫尔斯指数的新线性系统。我们将通过研究简化的线性哈密顿系统的莫尔斯指数来发展指数理论并定义相对莫尔斯指数。将指标理论与 Bartsch 和 Ding 开发的广义链接定理相结合 [不可度量化向量空间的变形定理及其在临界点理论中的应用。数学 Nachr。2006;279:1267–1288.]，获得了渐近二次非线性解的存在性和多样性。