Hidden symmetries in non-self-adjoint graphs

Abstract

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, Krej\v{c}i\v{r}\'{\i}k, 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 Schr\"odinger 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.