Related papers: Splitting Quantum Graphs
This note points out some bounds for the number of negative eigenvalues of Schroedinger operators with Hardy-type potentials, which follow from a simple coordinate transformation, and could prove useful in a spectral analysis of certain…
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…
Given an unbalanced open quantum graph, we derive a formula relating sums over its scattering resonances with integrals outside a strip. We deduce lower bounds on the number of resonances (in bounded regions of the complex plane),that are…
In this paper we obtain sharp Lieb-Thirring inequalities for a Schr\"odinger operator on semi-axis with a matrix potential and show how they can be used to other related problems. Among them are spectral inequalities on star graphs and…
We give general spectral and eigenvalue perturbation bounds for a selfadjoint operator perturbed in the sense of the pseudo-Friedrichs extension. We also give several generalisations of the aforementioned extension. The spectral bounds for…
Schr\"odinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely,…
We consider discrete Schr\"odinger operators with real periodic potentials on periodic graphs. The spectra of the operators consist of a finite number of bands. By "rolling up" a periodic graph along some appropriate directions we obtain…
In this work, we explore graph partitioning (GP) using quantum annealing on the D-Wave 2X machine. Motivated by a recently proposed graph-based electronic structure theory applied to quantum molecular dynamics (QMD) simulations, graph…
We present quantum complexity lower and upper bounds for independent set problems in graphs. In particular, we give quantum algorithms for computing a maximal and a maximum independent set in a graph. We present applications of these…
We prove quantum ergodicity for a family of periodic Schr\"odinger operators $H$ on periodic graphs. This means that most eigenfunctions of $H$ on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our…
Quantum image processing is a growing field attracting attention from both the quantum computing and image processing communities. We propose a novel method in combining a graph-theoretic approach for optimal surface segmentation and hybrid…
Efficient sampling from a classical Gibbs distribution is an important computational problem with applications ranging from statistical physics over Monte Carlo and optimization algorithms to machine learning. We introduce a family of…
We calculate statistical properties of the eigenfunctions of two quantum systems that exhibit intermediate spectral statistics: star graphs and Seba billiards. First, we show that these eigenfunctions are not quantum ergodic, and calculate…
We study Schr\"odinger operators on quantum graphs where the number of edges between points is determined by orbits of a "shift of finite type". We prove Anderson localization for these systems.
We study a graph partitioning problem motivated by the simulation of the physical movement of multi-body systems on an atomistic level, where the forces are calculated from a quantum mechanical description of the electrons. Several advanced…
We obtain a system of identities relating boundary coefficients and spectral data for the one-dimensional Schr\"{o}dinger equation with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter.…
A discrete Schr\"odinger operator of a graph $G$ is a real symmetric matrix whose $i,j$-entry, $i \neq j$, is negative if $\{i,j\}$ is an edge and zero if it is not an edge, while diagonal entries can be any real numbers. The discrete…
The purpose of this work is to study spectral methods to approximate the eigenvalues of nonlocal integral operators. Indeed, even if the spatial domain is an interval, it is very challenging to obtain closed analytical expressions for the…
In the first part of this manuscript a relationship between the spectrum of self-adjoint operator matrices and the spectra of their diagonal entries is found. This leads to enclosures for spectral points and in particular, enclosures for…
We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval $I$, in…