On a minimax principle in spectral gaps

On the minimax principle for Coulomb-Dirac operators

Let q and v be symmetric sesquilinear forms such that v is a form perturbation of q. Then we can associate a unique self-adjoint operator B to q+ v. Assuming that B has a gap (a, b) in the essential spectrum, we prove a minimax principle for the eigenvalues of B in (a, b) using a suitable orthogonal decomposition of the domain of q. This allows us to justify two minimax characterisations of eigenvalues in the gap of three-dimensional Dirac operators with electrostatic potentials having strong Coulomb singularities.

Minimax principles, Hardy-Dirac inequalities and operator cores for two and three dimensional Coulomb-Dirac operators

For $n\in\{2,3\}$ we prove minimax characterisations of eigenvalues in the gap of the $n$ dimensional Dirac operator with an potential, which may have a Coulomb singularity with a coupling constant up to the critical value $1/(4-n)$. This result implies a so-called Hardy-Dirac inequality, which can be used to define a distinguished self-adjoint extension of the Coulomb-Dirac operator defined on $\mathsf{C}_{0}^{\infty}(\mathbb{R}^n\setminus\{0\};\mathbb{C}^{2(n-1)})$, as long as the coupling constant does not exceed $1/(4-n)$. We also find an explicit description of an operator core of this op…

加权森林上 Cheeger $k$-常数的组合 Courant-Fischer-Weyl 极小极大原理

Combinatorial Courant-Fischer-Weyl Minimax Principle on Cheeger $k$-constants of Weighted Forests

本文为加权森林的 Cheeger $k$-常数建立组合极小极大原理，证明森林 1-Laplacian 特征值独立性，提出改进高阶 Cheeger 不等式，并给出 $h_k=λ_k(Δ_1)$ 的组合证明。

Complexified cones. Spectral gaps and variational principles

We consider contractions of complexified real cones, as recently introduced by Rugh in [Rugh10]. Dubois [Dub09] gave optimal conditions to determine if a matrix contracts a canonical complex cone. First we generalize his results to the case of complex operators on a Banach space and give precise conditions for the contraction and an improved estimate of the size of the associated spectral gap. We then prove a variational formula for the leading eigenvalue similar to the Collatz-Wielandt formula for a real cone contraction. Morally, both cases boil down to the study of suitable collections of 2…

Spectral gaps, additive energy, and a fractal uncertainty principle

We obtain an essential spectral gap for $n$-dimensional convex co-compact hyperbolic manifolds with the dimension $δ$ of the limit set close to $(n-1)/2$. The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors-David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.

谱隙中eigenvalue计数函数的算子理论方法

Operator theoretic methods for the eigenvalue counting function in spectral gaps

利用谱流的概念，我们提出了一个简单的方法来解决各种渐近问题，涉及特征值的间隙的本质谱的自伴算子。我们的方法使用了一些元素的谱移函数理论。使用这种方法，我们提供了概括和简化的证明，在这方面已经存在的文献中的两个结果。我们还给出了广义Birman-Schwinger原理的一个新的证明。