论文标题
在非自我接合图中的隐藏对称性
Hidden symmetries in non-self-adjoint graphs
论文作者
论文摘要
在有限的度量图上，研究了Laplace运算符的所有实现的集合。这些是通过在顶点的耦合边界条件来定义的，其中大多数定义了非自动接合运算符。在[Hussein，Krejčiči\'ık，siegl，trans。阿米尔。数学。 Soc。，367（4）：2921---2957，2015]通过提出了通过参数化矩阵的Cayley变换的边界条件规则性概念。这里介绍的要点是，不仅这种Cayley变换的存在对于基本光谱特性，而且对其杆和其渐近行为都是必不可少的。结果表明，这些极点和渐近学可以使用Quasi-Weierstrass正常形式来表征，该形式暴露了系统的某些“隐藏”对称性。因此，人们不仅可以分析这些主要是非自我的laplacians的光谱理论，还可以分析有限度量图上的时间依赖性热，波和schrödinger方程的良好性作为初始构造价值问题。特别是，可以表征$ C_0 $的发电机和分析半群和$ C_0 $ - cosine运算符功能。在星形图上，获得了有界$ C_0 $组的发电机的表征，因此获得了类似于自动偶会的运算符的表征。
On finite metric graphs the set of all realizations of the Laplace operator in the edgewise defined $L^2$-spaces are studied. These are defined by coupling boundary conditions at the vertices most of which define non-self-adjoint operators. In [Hussein, Krejčiř\'ık, Siegl, Trans. Amer. Math. Soc., 367(4):2921--2957, 2015] a notion of regularity of boundary conditions by means of the Cayley transform of the parametrizing matrices has been proposed. The main point presented here is that not only the existence of this Cayley transform is essential for basic spectral properties, but also its poles and its asymptotic behaviour. It is shown that these poles and asymptotics can be characterized using the quasi-Weierstrass normal form which exposes some "hidden" symmetries of the system. Thereby, one can analyse not only the spectral theory of these mostly non-self-adjoint Laplacians, but also the well-posedness of the time-dependent heat-, wave- and Schrödinger equations on finite metric graphs as initial-boundary value problems. In particular, the generators of $C_0$- and analytic semigroups and $C_0$-cosine operator functions can be characterized. On star-shaped graphs a characterization of generators of bounded $C_0$-groups and thus of operators similar to self-adjoint ones is obtained.
