相关论文: Landscape approximation of the ground state eigenv…
Let $D\subset R^d$ be a bounded domain and let $\mathcal P(D)$ denote the space of probability measures on $D$. Consider a Brownian motion in $D$ which is killed at the boundary and which, while alive, jumps instantaneously according to a…
The one-dimensional Hubbard model with different on-site interactions is investigated by renormalization group technique. In the case of a 1/4-filled band the dynamical nonequivalence of sites leads to the appearance of Umklapp processes in…
We study localization properties of low-lying eigenfunctions $$(-\Delta +V) \phi = \lambda \phi \qquad \mbox{in}~\Omega$$ for rapidly varying potentials $V$ in bounded domains $\Omega \subset \mathbb{R}^d$. Filoche & Mayboroda introduced…
The diagram technique for the one-band Hubbard model is formulated for the case of moderate to strong Hubbard repulsion. The expansion in powers of the hopping constant is expressed in terms of site cumulants of electron creation and…
Consider a random regular graph of fixed degree $d$ with $n$ vertices. We study spectral properties of the adjacency matrix and of random Schr\"odinger operators on such a graph as $n$ tends to infinity. We prove that the integrated density…
We present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs. There is a natural conjecture arising from analogy with random matrix theory that connects these spectra…
We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form $-{div} (A\nabla u) + Vu + f(u^2) u = \lambda u$, $\|u\|_{L^2}=1$. We…
We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower…
We discuss Laplacian spectrum on a finite metric graph with vertex couplings violating the time-reversal invariance. For the class of star graphs we determine, under the condition of a fixed total edge length, the configurations for which…
We derive the analytical expression of the ground state of the Hubbard model with unconstrained hopping at half filling and for arbitrary lattice sites.
In this work we analyze basic properties of Random Apollonian Networks \cite{zhang,zhou}, a popular stochastic model which generates planar graphs with power law properties. Specifically, let $k$ be a constant and $\Delta_1 \geq \Delta_2…
Spectral methods include a family of algorithms related to the eigenvectors of certain data-generated matrices. In this work, we are interested in studying the geometric landscape of the eigendecomposition problem in various spectral…
We develop a linear theory of very weak solutions for nonlocal eigenvalue problems $\mathcal L u = \lambda u + f$ involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and…
We investigate the ground-state properties of the two-dimensional Hubbard model, based on the off-diagonal wave function variational Monte Carlo method. We use an optimized wave function that is improved from an initial one-body wave…
We present rigorous results for the SU($n$) Fermi-Hubbard models with finite-range hopping in $d$ ($\ge 2$) dimensions. The models are defined on a class of decorated lattices. We first study the models with flat bands at the bottom of the…
The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph,…
We investigate the stability domains of ground states of generalized Hubbard models with next-nearest neighbour interaction using the optimum groundstate approach. We focus on the $\eta$-pairing state with momentum P=0 and the fully…
We study the behaviour of the first eigenfunction of the Dirichlet Laplacian on a planar convex domain near its maximum. We show that the eccentricity and orientation of the superlevel sets of the eigenfunction stabilise as they approach…
We use probabilistic tools based on Brownian motion and Feynman-Kac formulae to investigate the heat profile for the ground state Dirichlet and second Neumann eigenfunctions. Among other topics, we comment on supremum norm bounds for ground…
We study how the optimization landscape of a neural network deforms as a non-linear activation is introduced through a smooth homotopy. Working first in an abstract local setting - a smooth one-parameter family of objective functions…