We interpret the global Hadamard condition for a two-point distribution of a Klein-Gordon neutral scalar quantum field model on an arbitrary globally hyperbolic curved space-time in terms of distinguished parametrices (of Duistermaat and Hörmander) and a wave front set spectrum condition. Microlocal results by Duistermaat and Hörmander such as the propagation of singularities theorem and the un...

A fundamental and challenging problem in spectral graph theory is to characterize which graphs are uniquely determined by their spectra. In Wang (2017), the author proved that an n -vertex G its generalized spectrum (DGS) whenever 2 ? ? ? det W odd square-free. Here, walk matrix of , namely, = [ e … 1 ] with all-one vector adjacency . this paper, we focus on a larger family d square-free, where...

The condition on the potential ensuring the discreteness of the spectrum of infinite quantum graph Hamiltonian is considered. The corresponding necessary and sufficient condition was obtained by Molchanov 50 years ago. In the present paper, the analogous condition is obtained for a quantum graph. A quantum graph with infinite leads (edges) or/and infinite chains of vertices such that neighbor o...