Related papers: A Gelfand-Levitan trace formula for generic quantu…
We look at the various aspects of treating general relativity as a quantum theory. It is briefly studied how to consistently quantize general relativity as an effective field theory. A key achievement here is the long-range low-energy…
Eigenvalue behavior for the equation -\lambda y"=Vu on the edges of a graph G of final total length, with a non-negative weight function V and under the Kirchhoff matching conditions at the vertices and zero boundary condition at at least…
We study the statistical properties of the scattering matrix associated with generic quantum graphs. The scattering matrix is the quantum analogue of the classical evolution operator on the graph. For the energy-averaged spectral form…
The classical concept of $Q$-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be…
We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As…
For self-similar measures with overlaps, closed formulas of the $L^q$-spectrum have been obtained by Ngai and the author for measures that are essentially of finite type in [J. Aust. Math. Soc. \textbf{106} (2019), 56--103]. We extend the…
A voltage graph is a finite directed graph whose edges are labeled by elements of a finite group $G$. A classical construction of Gross and Tucker associates to every voltage graph with vertex set $V$ a so-called derived graph with vertex…
Trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs. The basis of our work is the result by Kurasov which links the Euler characteristic $\chi$ of metric graphs to the spectrum of their…
Let $\mbox{odd}(G)$ and $i(G)$ denote the number of nontrivial odd components and the number of isolated vertices of a graph $G$, respectively. The $k$-Berge-Tutte-formula of a graph $G$ is defined as:…
We consider quantum search on graphs. Recently, it has been shown that the graph properties like connectivity, global symmetry, or regularity cannot serve as a reliable criteria that must be satisfied by a graph to allow a successful…
We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs enjoys a rich group of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the…
We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups, called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state…
A semiclassical approach to the universal ergodic spectral statistics in quantum star graphs is presented for all known ten symmetry classes of quantum systems. The approach is based on periodic orbit theory, the exact semiclassical trace…
Using Cartan's equivalence method for point transformations we obtain from first principles the conformal geometry associated with third order ODEs and a special class of PDEs in two dimensions. We explicitly construct the null tetrads of a…
Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in…
A new generalized formulation of the spectral condition is proposed for quantum fields with highly singular infrared behavior whose vacuum correlation functions are well defined only under smearing with analytic test functions in momentum…
We pick up a solvable ${\cal PT}-$symmetric quantum square well on an interval of $x \in := (-L,L)\mathbb{G}^{(2)}$ (with an $\alpha-$dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just…
We consider a so-called quantum graph with standard continuity and Kirchhoff vertex conditions where the Kirchhoff vertex condition is perturbed by Gaussian noise. We show that the quantum graph setting is very different from the classical…
This paper is a mathematical study of quantum correlation functions in quantum field theory within a homotopy algebraic framework motivated from the BV quantization scheme. We characterize quantum correlation functions by algebraic homotopy…
An ultragraph gives rise to a labelled graph with some particular properties. In this paper we describe the algebras associated to such labelled graphs as groupoid algebras. More precisely, we show that the known groupoid algebra…