Abstract

On finite metric graphs the set of all realizations of the Laplace operator in the edgewise defined L²-spaces are studied. These are defined by coupling boundary conditions at the vertices most of which define non-self-adjoint operators. In [Hussein, Krejčiřík, 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 behavior. 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 analyze 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 C₀- and analytic semigroups and C₀-cosine operator functions can be characterized. On star-shaped graphs a characterization of generators of bounded C₀-groups and thus of operators similar to self-adjoint ones is obtained.

摘要

在有限度量图上，研究了边定义的L ²空间中拉普拉斯算子的所有实现的集合。这些由顶点处的耦合边界条件定义，其中大多数定义了非自伴随算子。在 [侯赛因，Krejčiřík，Siegl，Trans。阿米尔。数学。社会。, 367(4):2921–2957, 2015] 通过参数化矩阵的凯莱变换提出了边界条件规则性的概念。这里提出的要点是，不仅这个 Cayley 变换的存在对于基本谱属性来说是必不可少的，而且它的极点和它的渐近行为也是必不可少的。结果表明，这些极点和渐近线可以使用准魏尔斯特拉斯范式来表征，该范式暴露了系统的一些“隐藏”对称性。因此，人们不仅可以分析这些主要是非自伴随拉普拉斯算子的谱理论，还可以分析作为初始边界值问题的有限度量图上的瞬态热、波和薛定谔方程的适定性。特别地，C ₀ - 和解析半群的生成元和可以表征C _{0 -}余弦算子函数。在星形图上，获得了有界C ₀ -群的生成器的特征，从而获得了类似于自伴随的算子的特征。