Let (M, g, σ) be a compact Riemannian surface equipped with a spin structure σ. For any metric ḡ onM , we denote by ΣḡM the spinor bundle associated to ḡ. We let ∆ḡ be the Laplace-Beltrami operator acting on smooth functions of M and Dḡ be the Dirac operator acting on smooth spinor fields with respect to the metric ḡ. We also denote by μ1(ḡ) (resp. λ1(ḡ)) the smallest positive eigenvalue of ∆ḡ ...

First order perturbation theory for eigenvalues of arbitrary matrices is systematically developed in all its generality with the aid of the Newton diagram, an elementary geometric construction first proposed by Isaac Newton. In its simplest form, a square matrix A with known Jordan canonical form is linearly perturbed to A(ε) = A+ ε B for an arbitrary perturbation matrix B, and one is intereste...

We estimate the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature in terms of the diameter and the lower bound of Ricci curvature and give an affirmative answer to the conjecture of H. C. Yang.

We prove some results about the first Steklov eigenvalue d1 of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality [9] may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d1 for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problem...

This paper discusses finite-element highly efficient calculation schemes for solving eigenvalue problem of electric field. Multigrid discretization is extended to the filter approach for eigenvalue problem of electric field. With this scheme one solves an eigenvalue problem on a coarse grid just at the first step, and then always solves a linear algebraic system on finer and finer grids. Theore...

We prove the existence of a spectral resolution of the Wheeler-DeWitt equation when the matter field is provided by a massive Yang-Mills field. The resolution is achieved by first solving the free eigenvalue problem for the gravitational field and then the constrained eigenvalue problem for the Yang-Mills field. In the latter case the mass of the Yang-Mills field assumes the role of the eigenva...

Trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence are characterized. It is shown that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p.

Let M be a compact manifold with a spin structure χ and a Riemannian metric g. Let λ2g be the smallest eigenvalue of the square of the Dirac operator with respect to g and χ. The τ -invariant is defined as τ(M,χ) := sup inf q λ2gVol(M, g) 1/n where the supremum runs over the set of all conformal classes on M , and where the infimum runs over all metrics in the given class. We show that τ(T , χ)...

We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich’s eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.