The main object of this paper is to study the essential spectrum of a Hamiltonian system of arbitrary order with one singular endpoint under a class of perturbations. We first present a characterization of the essential spectrum in terms of singular sequences and then give the concept of perturbations small at singular endpoints of Hamiltonian systems. Based on the above characterization, the invariance of essential spectra of Hamiltonian systems under these perturbations is shown. It is noted that these perturbations are given by using the associated pre-minimal operator , which provides great convenience in the study of essential spectra of Hamiltonian systems since each element of the domain of has compact support. As applications, some sufficient conditions for the invariance of essential spectra of some systems are obtained in terms of coefficients of systems and perturbations terms. Further, essential spectra of Hamiltonian systems with different weight functions are discussed. Here, Hamiltonian systems may be non-symmetric.

中文翻译：

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一类扰动下任意阶奇异哈密顿微分算子的基本谱

本文的主要目的是研究一类扰动下具有一个奇异端点的任意阶哈密顿系统的本质谱。我们首先根据奇异序列给出基本谱的表征，然后给出哈密顿系统奇异端点处微扰的概念。基于上述特征，给出了这些扰动下哈密顿系统本质谱的不变性。注意到这些扰动是通过使用相关的前极小算子 给出的，这为研究哈密顿系统的本质谱提供了很大的便利，因为域的每个元素有紧凑的支持。作为应用，从系统系数和微扰项中得到了一些系统本质谱不变性的充分条件。此外，讨论了具有不同权函数的哈密顿系统的基本谱。在这里，哈密顿系统可能是非对称的。