We consider the problem of reconstructing an $n\times n$ cell matrix $D\left(\overrightarrow{x}\right)$ constructed from a vector $\overrightarrow{x}=(x_1,x_2,\cdots ,x_n)$ of positive real numbers, from a given set of spectral data. In addition, we show that the spectra of cell matrices $D\left(\overrightarrow{x}\right)$ and $D\left(\pi \left(\overrightarrow{x}\right)\right)$ are the same for every permutation $\pi \in S_n$.

The Hartree-Fock exchange operator is an integral operator arising in the Hartree-Fock model as well as in some instances of the density functional theory. In a number of applications, it is convenient to approximate this integral operator by a multiplication operator, i.e. by a local potential. This article presents a detailed analysis of the mathematical properties of various local approximations to the nonlocal Hartree-Fock exchange operator including the Slater potential, the optimized effective...

Within the effective mass and nonparabolic band theory, a general framework of mathematical models and numerical methods is developed for theoretical studies of semiconductor quantum dots. It includes single-electron models and many-electron models of Hartree-Fock, configuration interaction, and current-spin density functional theory approaches. These models result in nonlinear eigenvalue problems from a suitable discretization. Cubic and quintic Jacobi-Davidson methods of block or nonblock version...

We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when $\lambda \to {\lambda }_1$ from the left, ${\lambda }_1$ being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.

We obtain a non-existence result for a class of quasi-linear eigenvalue problems when a parameter is small. By using Pohozaev identity and some comparison arguments, non-existence theorems are established for quasi-linear eigenvalue problems under supercritical growth condition.

The nonlinear eigenvalue problem for p-Laplacian $$\left\{\begin{array}{c}-{div\left(a\left(x\right)\right|\nabla u|}^{p-2}{\nabla u)=\lambda g\left(x\right)|u|}^{p-2}u\text{in}{ℝ}^N,u>0\text{in}{ℝ}^N,\underset{\left|x\right|\to \infty }{lim}u\left(x\right)=0,\hfill \end{array}\right.$$ is considered. We assume that $1<p<N$ and that $g$ is indefinite weight function. The existence and $C^{1,\alpha }$-regularity of the weak solution is proved.

The non-local Gel’fand problem, $\Delta v+\lambda e^v/{\int }_{\Omega }{e}^{v}dx=0$ with Dirichlet boundary condition, is studied on an n-dimensional bounded domain Ω. If it is star-shaped, then we have an upper bound of λ for the existence of the solution. We also have infinitely many bendings in λ of the connected component of the solution set in λ,v if Ω is a ball and 3 ≤ n ≤ 9.