论文标题
常规线性非汉顿系统的重新归一化振荡理论
Renormalized oscillation theory for regular linear non-Hamiltonian systems
论文作者
论文摘要
在最近的工作中，Baird等人。已将马斯洛夫指数的定义概括为格拉曼尼亚子空间的路径，而拉格朗日格拉马尼亚人不一定包含[T. T. J. Baird，P。Cornwell，G。Cox，C。Jones和R. Marangell，{\ It Maslov Non-Hamiltonian Systems的概括性指数}，Siam J. Math。肛门。 {\ bf 54}（2022）1623-1668]。这样的扩展为非黑米顿系统的应用程序开辟了可能性，而Baird及其合作者利用了这一观察结果来建立振荡型结果，以在此广义环境中获得特征值计数的较低界限。在当前的分析中，作者表明，在这种广义环境中适当定义的重新归一化的振荡理论可以自然地应用，并且它具有优势，因为在传统的线性汉密尔顿系统的环境中，可以确保交叉点的单调性作为独立变量的单调性，以增加系统/边界条件组合的广泛范围。这似乎标志着将重新归一化振荡方法扩展到非汉密尔顿环境的第一个努力。
In recent work, Baird et al. have generalized the definition of the Maslov index to paths of Grassmannian subspaces that are not necessarily contained in the Lagrangian Grassmannian [T. J. Baird, P. Cornwell, G. Cox, C. Jones, and R. Marangell, {\it Generalized Maslov indices for non-Hamiltonian systems}, SIAM J. Math. Anal. {\bf 54} (2022) 1623-1668]. Such an extension opens up the possibility of applications to non-Hamiltonian systems of ODE, and Baird and his collaborators have taken advantage of this observation to establish oscillation-type results for obtaining lower bounds on eigenvalue counts in this generalized setting. In the current analysis, the author shows that renormalized oscillation theory, appropriately defined in this generalized setting, can be applied in a natural way, and that it has the advantage, as in the traditional setting of linear Hamiltonian systems, of ensuring monotonicity of crossing points as the independent variable increases for a wide range of system/boundary-condition combinations. This seems to mark the first effort to extend the renormalized oscillation approach to the non-Hamiltonian setting.
