Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpectral estimates for the Heisenberg Laplacian on cylinders
289
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study Riesz means of eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on a cylinder in dimension three. We obtain an inequality with a sharp leading term and an additional lower order term.
rate research
Read More
We consider the operator $H={d^4dt^4}+{ddt}p{ddt}+q$ with 1-periodic coefficients on the real line. The spectrum of $H$ is absolutely continuous and consists of intervals separated by gaps. We describe the spectrum of this operator in terms of the Lyapunov function, which is analytic on a two-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. We describe the spectrum of $H$ in terms of periodic, antiperiodic eigenvalues, and so-called resonances. We prove that 1) the spectrum of $H$ at high energy has multiplicity two, 2) the asymptotics of the periodic, antiperiodic eigenvalues and of the resonances are determined at high energy, 3) for some specific $p$ the spectrum of $H$ has an infinite number of gaps, 4) the spectrum of $H$ has small spectral band (near the beginner of the spectrum) with multiplicity 4 and its asymptotics are determined as $pto 0, q=0$.
This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity) and its associated eigenvector, the so-called Fiedler vector. Here we prove that, given a Laplacian matrix, it is possible to perturb the weights of the existing edges in the underlying graph in order to obtain simple eigenvalues and a Fiedler vector composed of only non-zero entries. These structural genericity properties with the constraint of not adding edges in the underlying graph are stronger than the classical ones, for which arbitrary structural perturbations are allowed. These results open the opportunity to understand the impact of structural changes on the dynamics of complex systems.
We consider the inverse scattering on the quantum graph associated with the hexagonal lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the S-matrix for all energies in any given open set in the continuous spectrum determines the potentials.
Given two Hilbert spaces, $mathcal{H}$ and $mathcal{K}$, we introduce an abstract unitary operator $U$ on $mathcal{H}$ and its discriminant $T$ on $mathcal{K}$ induced by a coisometry from $mathcal{H}$ to $mathcal{K}$ and a unitary involution on $mathcal{H}$. In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpinski lattice, which is pure point and has a Cantor-like structure.
We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({mathbb R^2})$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $Lambda_q$, $q in {mathbb Z}_+$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${rm Op}^{rm w}({mathcal V})$ with real-valued Weyl symbol ${mathcal V}$, such that ${rm Op}^{rm w}({mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{{mathcal V}} = H_0 + {rm Op}^{rm w}({mathcal V})$. First, we construct symbols ${mathcal V}$, possessing a suitable symmetry, such that the operator $H_{mathcal V}$ admits an explicit eigenbasis in $L^2({mathbb R^2})$, and calculate the corresponding eigenvalues. Moreover, for ${mathcal V}$ which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of $H_{mathcal V}$ adjoining any given $Lambda_q$. We find that the effective Hamiltonian in this context is the Toeplitz operator ${mathcal T}_q({mathcal V}) = p_q {rm Op}^{rm w}({mathcal V}) p_q$, where $p_q$ is the orthogonal projection onto ${rm Ker}(H_0 - Lambda_q I)$, and investigate its spectral asymptotics.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
