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We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we…
We study Hamiltonians with point interactions in spaces of vector-valued functions. Using some information from the theory of quantum graphs we describe a class of the operators which can be reduced to the direct sum of several…
We consider the Schroedinger operator on graphs and study the spectral statistics of a unitary operator which represents the quantum evolution, or a quantum map on the graph. This operator is the quantum analogue of the classical evolution…
On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity…
We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma)$ in $L^2(\mathbb{R}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an…
We consider rectangular graph superlattices of sides l1, l2 with the wavefunction coupling at the junctions either of the delta type, when they are continuous and the sum of their derivatives is proportional to the common value at the…
Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. An infinite series of trace formulae is obtained which link together two…
We describe the boundary conditions at the vertex that one must choose to obtain a dynamical system that best describes the low-energy part of the evolution of a quantum system confined to a very small neighbourhood of a star-shaped metric…
We study Schr\"odinger operators on compact finite metric graphs subject to $\delta'$-coupling conditions. Based on a novel modified local Weyl law, we derive an explicit expression for the limiting mean eigenvalue distance of two different…
We study a set of scattering matrices of quantum graphs containing minimal number of passbands, i.e., maximal number of zero elements. The cases of even and odd vertex degree are considered. Using a solution of inverse scattering problem,…
We consider a family of open sets $M_\epsilon$ which shrinks with respect to an appropriate parameter $\epsilon$ to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted…
We put into evidence graphs with adjacency operator whose singular subspace is prescribed by the kernel of an auxiliary operator. In particular, for a family of graphs called admissible, the singular continuous spectrum is absent and there…
We study Schr\"odinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by $\delta$-couplings with a parameter $\alpha\in\R$. If the graph is "straight", i.e. periodic…
We construct rich families of Schr\"odinger operators on symmetric graphs, both quantum and combinatorial, whose spectral degeneracies are persistently larger than the maximal dimension of an irreducible representations of the symmetry…
In this note we sharpen the lower bound from [LLP10] on the spectrum of the 2D Schroedinger operator with a delta-interaction supported on a planar angle. Using the same method we obtain the lower bound on the spectrum of the 2D…
This paper deals with the approximation of a magnetic Schr\"odinger operator with a singular $\delta$-potential that is formally given by $(i \nabla + A)^2 + Q + \alpha \delta_\Sigma$ by Schr\"odinger operators with regular potentials in…
We prove some abstract Wegner bounds for random self-adjoint operators. Applications include elementary proofs of Wegner estimates for discrete and continuous Anderson Hamiltonians with possibly sparse potentials, as well as Wegner bounds…
We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms…
We show that a Schr\"odinger operator $A_{\delta, \alpha}$ with a $\delta$-interaction of strength $\alpha$ supported on a bounded or unbounded $C^2$-hypersurface $\Sigma \subset \mathbb{R}^d$, $d\ge 2$, can be approximated in the norm…
We consider discrete Schr\"odinger operators $H_{\mu Q}=\Delta+\mu Q$ with real periodic potentials $Q$ on periodic graphs, where $\Delta$ is the adjacency operator and $\mu\in\mathbb R$ is a coupling constant. The spectra of the operators…