Related papers: Spectral determinant on graphs with generalized bo…

The aim of this article is to present a complete system of Floquet spectral invariants for the discrete Schr\"odinger operators with periodic potentials on periodic graphs. These invariants are polynomials in the potential and determined by…

We prove sharp upper bounds for eigenvalues of Schr\"odinger operators on quantum graphs with $\delta$-coupling (also known as Robin) conditions at all vertices. The bounds depend on the geometry of the graph, on the potential, and the…

Let $H_V=-\Delta +V$ be a Schr\"odinger operator on an arbitrary open set $\Omega$ of $\mathbb R^d$, where $d \geq 3$, and $\Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $\Omega$. The purpose of this…

We consider continuum random Schr\"odinger operators of the type $H_{\omega} = -\Delta + V_0 + V_{\omega}$ with a deterministic background potential $V_0$. We establish criteria for the absence of continuous and absolutely continuous…

We consider the spin-$J$ XXZ-Hamiltonian on general graphs $\mathcal{G}$ and show its equivalence to a direct sum of discrete many-particle Schr\"odinger type operators on what we call "$N$-particle graphs with maximal local occupation…

In the first part of the paper we consider the Schr\"odinger operator $ -\Delta-V(x),\quad V>0. $ We discuss the relation between the behavior of $V$ at the infinity and the properties of the negative spectrum of $H$. After that, we…

We consider the scattering theory for the Schr\"odinger operator $-\Dc_x^2+V(x)$ on graphs made of one-dimensional wires connected to external leads. We derive two expressions for the scattering matrix on arbitrary graphs. One involves…

This paper is concerned with the discrete spectra of Schroedinger operators H = -Delta + V, where V(r) is an attractive potential in N spatial dimensions. Two principal results are reported for the bottom of the spectrum of H in each…

We study Schr\"{o}dinger operators on star metric graphs with potentials of the form $\alpha\varepsilon^{-2}Q(\varepsilon^{-1}x)$. In dimension 1 such potentials, with additional assumptions on $Q$, approximate in the sense of distributions…

For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on the graph only through its degree and its…

We consider the Schr\"odinger operator $$-\frac{d^2}{d x^2} + V \qquad \mbox{on an interval}~~[a,b]~\mbox{with Dirichlet boundary conditions},$$ where $V$ is bounded from below and prove a lower bound on the first eigenvalue $\lambda_1$ in…

The paper deals with singular Schr\"odinger operators of the form \begin{gather*} -{\mathrm{d}^2\over \mathrm{d} x^2 } + \sum_{k\in\mathbb{Z} }\gamma_k \delta(\cdot-z_k),\quad \gamma_k\in\mathbb{R}, \end{gather*} in…

The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and…

The behavior of the discrete spectrum of the Schr\"odinger operator $-\D - V$, in quite a general setting, up to a large extent is determined by the behavior of the corresponding heat kernel $P(t;x,y)$ as $t\to 0$ and $t\to\infty$. If this…

We consider the multidimensional Borg-Levinson theorem of determining both the magnetic field $dA$ and the electric potential $V$, appearing in the Dirichlet realization of the magnetic Schr\"odinger operator $H=(-{\rm i}\nabla+A)^2+V$ on a…

Given a potential $V$ and the associated Schr\"odinger operator $-\Delta+V$, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example $V$ or $V^{-1}$ enjoys suitable…

The paper studies the spectral properties of the Schr\"odinger operator $A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the free Laplacian…

We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schroedinger operator on a star-shaped…

We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows "forbidden" endpoints and permits the consideration of functions exhibiting very limited regularity. This is…

We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on…