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Energy functionals of the Green's function can simultaneously provide spectral and thermodynamic properties of interacting electrons' systems. Though powerful in principle, these formulations need to deal with dynamical…
Here we review the many aspects and distinct phenomena associated to quantum dynamics on general graph structures. For so, we discuss such class of systems under the energy domain Green's function ($G$) framework. This approach is…
Consider a sequence of finite regular graphs (GN) converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero…
We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…
The quantum correlations of $N$ noninteracting spinless fermions in their ground state can be expressed in terms of a two-point function called the kernel. Here we develop a general and compact method for computing the kernel in a general…
Quantum properties of the state associated to the gluon Green's function in the BFKL approach are studied using a discretization in virtuality space. Considering the coupling constant as imaginary, its density matrix corresponds to a pure…
Quantum graphity is a background independent model for emergent geometry, in which space is represented as a complete graph. The high-energy pre-geometric starting point of the model is usually considered to be the complete graph, however…
We establish the $\Gamma$-convergence of some energy functionals describing nonlocal attractive interactions in bounded domains. The interaction potential solves an elliptic equation (local or nonlocal) in the bounded domain and the primary…
Low-energy continuous states of electron in heterosrtucture with periodically placed quantum-dot sheets are studied theoretically. The Green's function of electron is governed by the Dyson equation with the self-energy function which is…
We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed…
In this thesis, we study Laplacian eigenfunctions on metric graphs, also known as quantum graphs. We restrict the discussion to standard quantum graphs. These are finite connected metric graphs with functions that satisfy Neumann vertex…
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…
We study the ground-state properties of a double layer graphene system with the Coulomb interlayer electron-electron interaction modeled within the random phase approximation. We first obtain an expression of the quantum capacitance of a…
We address the problem of interacting electrons in an external potential by introducing the occupied spectral density $\rho(\mathbf{r},\omega)$ as fundamental variable. First, we formulate the problem using an embedding framework, and prove…
We expand the quantum mechanical wavefunction in a complete set of square integrable orthonormal basis such that the matrix representation of the Hamiltonian operator is tridiagonal and symmetric. Consequently, the matrix wave equation…
By computing the local energy expectation values with respect to some local measurement basis we show that for any quantum system there are two fundamentally different contributions: changes in energy that do not alter the local von Neumann…
We normalize the combinatorial Laplacian of a graph by the degree sum, look at its eigenvalues as a probability distribution and then study its Shannon entropy. Equivalently, we represent a graph with a quantum mechanical state and study…
We study the energy levels of graphene magnetic circular quantum dot surrounded by an infinite graphene sheet in the presence of an electrostatic potential. We solve Dirac equation to derive the solutions of energy spectrum associated with…
The response of an arbitrary scattering problem to quasi-static perturbations in the scattering potential is naturally expressed in terms of a set of local partial densities of states and a set of sensitivities each associated with one…
The spectral density of bound pairs in ideal 1D, 2D and Bethe lattices is computed for weak and strong interactions. The computations are performed with Green's functions by an efficient recursion method in real space. For the range of…