We review some basic ideas from the theory of overfilling families and use the special case of coherent states to prove Weyl asymptotics for the euclidean Laplacian and the hyperbolic Laplacian on a domain with Dirichlet boundary conditions.
Rebar cages are the skeletons of reinforced concrete structures. These temporary structures consist of longitudinal and transverse reinforcing bars connected by tie-wires. Given the relatively low strength of tie-wire connections, replacing tie-wires with mechanical connectors such as U-bolts can improve the stability and strength of rebar cages. This paper aims to understand the behavior of large prefabricated rebar cages reinforced with mechanical U-bolt connectors through experimental and analytical investigations. Twenty-six full-scale experimental tests are performed on five different underground pile-shaft rebar cages with tie-wire and U-bolt connectors to determine their behavior during different site handling conditions. The data obtained from the experiments are used to develop and calibrate detailed finite element models that can predict the complex response behavior of rebar cages reinforced with U-bolt connectors. The results show that U-bolt connectors can effectively ensure rebar cage integrity even under extreme loading conditions. Additionally, it is concluded that the presence of U-bolt connectors allows for the elimination of internal stiffening elements, which are common for large rebar cages, adding to the simplicity and efficiency of the construction process. The results of this study can be used as a basis for establishing analysis, design, fabrication, and handling guidelines for rebar cages.
The global structure of the spectrum of periodic non-Hermitian Jacobi operators is described by the discriminant and its stationary points. We also give necessary and sufficient conditions for real spectrum and single interval spectrum.
We improve the resolvent estimate in the Kreiss matrix theorem for a set of matrices that generate uniformly bounded semigroups. The new resolvent estimate is proved to be equivalent to Kreiss's resolvent condition, and it better describes the behavior of the resolvents at infinity.
In this work we consider the Anderson model on Bethe lattice and prove that the integrated density of states (IDS) is as smooth as the single site distribution (SSD), in high disorder
A KdV flow is constructed on a space whose structure is described in terms of the spectrum of the underlying Schrödinger operators. The space includes the conventional decaying functions and ergodic ones. Especially any smooth almost periodic function can be initial data for the KdV equation.
In this study, singular diffusion operator with jump conditions is considered. Integral representations have been derived for solutions that satisfy boundary conditions and jump conditions. Some properties of eigenvalues and eigenfunctions are investigated. Asymtotic representation of eigenvalues and eigenfunction have been obtained. Reconstruction of the singular diffusion operator have been shown by the Weyl function.
For the almost Mathieu operator with a small coupling constant, for a series of spectral gaps, we describe the asymptotic locations of the gaps and get lower bounds for their lengths. The results are obtained by analysing a monodromy matrix.
We investigate the low-energy behavior of the resolvent of Schrodinger operators with finitely many point interactions in three dimensions. We also discuss the occurrence and the multiplicity of zero energy obstructions.
We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We give necessary conditions for this Laplacian to be sectorial. We introduce a special self-adjoint operator and compare its essential spectrum with that of the non self-adjoint Laplacian considered.
A bounded operator $T$ in a Banach space $X$ is said to satisfy the essential descent spectrum equality, if the descent spectrum of $T$ as an operator on $X$ coincides with the essential descent spectrum of $T$. In this note, we give some conditions under which the equality $σ_{desc}(T) = σ^e_{desc}(T)$ holds for a single operator $T$.
We consider second-order functional differential operators with a constant delay. Properties of their spectral characteristics are obtained and a nonlinear inverse problem is studied, which consists in recovering the operators from their spectra. We establish the uniqueness and develop a constructive algorithm for solution of the inverse problem.
In this paper, we show that every pseudo B-Fredholm operator is a pseudo Fredholm operator. Afterwards, we prove that the pseudo B-Weyl spectrum is empty if and only if the pseudo B-Fredholm spectrum is empty. Also, we study a symmetric difference between some parts of the spectrum.
This text is a slightly expanded version of my 6 hour mini-course at the PDE-meeting in Évian-les-Bains in June 2009. The first part gives some old and recent results on non-self-adjoint differential operators. The second part is devoted to recent results about Weyl distribution of eigenvalues of elliptic operators with small random perturbations.