非自伴图的隐对称性
Hidden symmetries in non-self-adjoint graphs
摘要：
在有限度量图上，研究了Laplace算子在边定义的$L^2$ -空间中的所有实现的集合.这些定义的耦合边界条件的顶点，其中大多数定义非自伴算子。在[Hussein，Krejčisiness\'ık，Siegl，Trans. Amer. Math. Soc.，367（4）：2921--2957，2015]已经提出了借助于参数化矩阵的Cayley变换的边界条件的正则性的概念。这里提出的主要观点是，不仅存在这个凯莱变换是必不可少的基本谱性质，而且其极点和渐近行为。结果表明，这些极点和渐近的特点，可以使用准Weierstrass正常的形式，暴露了一些“隐藏”的对称性的系统。因此，人们不仅可以分析这些大多数非自伴拉普拉斯算子的谱理论，而且还可以分析有限度量图上的含时热方程、波动方程和薛定谔方程的适定性作为初边值问题。特别地，可以刻画$C_0$ -和解析半群以及$C_0$ -余弦算子函数的生成元.在星形图的有界$C_0$ -群的生成元的一个特征，从而类似于自伴算子。
英文摘要：
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.
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