Related papers: On the eigenvalue spacing distribution for a point…
In this paper, we consider the problem of the scattering of in-plane waves at an interface between a homogeneous medium and a metamaterial. The relevant eigenmodes in the two regions are calculated by solving a recently described non…
This work deals with the isogeometric Galerkin discretization of the eigenvalue problem related to the Laplace operator subject to homogeneous Dirichlet boundary conditions on bounded intervals. This paper uses GLT theory to study the…
A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator,…
We study the small eigenvalues of the Hodge Laplacian on collaping torus bundles with bounded curvature. In the first part of this dissertation, we consider examples of bundles on S^1 and T^2 with homogeneous structure. In the second part,…
The statistical properties of level spacings provide valuable insights into the dynamical properties of a many-body quantum systems. We investigate the level statistics of the Fermi-Hubbard model with dimerized hopping amplitude and find…
We consider the Landau Hamiltonian perturbed by a long-range electric potential $V$. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we obtain an estimate of the rate of…
The full spectrum of transfer matrices of the general eight-vertex model on a square lattice is obtained by numerical diagonalization. The eigenvalue spacing distribution and the spectral rigidity are analyzed. In non-integrable regimes we…
New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can be viewed as a discretized two point…
We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of…
We study the asymptotic behaviour of the eigenvalues of the Laplace-Beltrami operator on a compact hypersurface in \mathds{R}^{n+1} as it is flattened into a singular double-sided flat hypersurface. We show that the limit spectral problem…
We compute the asymptotic eigenvalue distribution of the neural tangent kernel of a two-layer neural network under a specific scaling of dimension. Namely, if $X\in\mathbb{R}^{n\times d}$ is an i.i.d random matrix, $W\in\mathbb{R}^{d\times…
The theory of random matrices with eigenvalues distributed in the complex plane and more general "beta-ensembles" (logarithmic gases in 2D) is reviewed. The distribution and correlations of the eigenvalues are investigated in the large N…
In the present work, eigenvalue distributions defined by a random rectangular matrix whose components are neither independently nor identically distributed are analyzed using replica analysis and belief propagation. In particular, we…
We describe our current understanding on the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which we previously observed through our numerical experiments. The eigenvalue…
We consider the spectrum of the discrete Schr\"odinger equation with one-dimensional perturbation. We obtain the explicit form of scattering matrix and find the exact condition of absence of singular part of the spectrum. We calculated also…
We study the spectrum of the Dirichlet Laplacian operator in a two-dimensional twisted strip embedded in $\mathbb R^d$ with $d \geq 2$. It is shown that a local twisting perturbation can create discrete eigenvalues for the operator. In…
The entanglement spectrum, i.e., the full distribution of Schmidt eigenvalues of the reduced density matrix, contains more information than the conventional entanglement entropy and has been studied recently in several many-particle…
On the unit interval (I), and the Sierpinski Gasket ($\mathcal{SG}$), the spectral decimation function of the Laplacian has similar properties that result in positive minimum spacing of eigenvalues. Other fractals, for example the level-3…
We study the probability distribution of the ratio of consecutive level spacings for embedded one plus two-body random matrix ensembles with and without spin degree of freedom and for both fermion and boson systems. The agreement between…
In this tutorial paper, we consider the problem of electromagnetic scattering by a bounded two-dimensional dielectric object, and discuss certain interesting properties of the scattered field. Using the electric field integral equation,…