Related papers: Eigenvalues of the basic Dirac operator on quatern…
For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number…
We review some recent results concerning lower eigenvalues estimates for the Dirac operator [6, 7]. We show that Friedrich's inequality can be improved via certain well-chosen symmetric tensors and provide an application to Sasakian spin…
In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac…
We obtain upper bounds for the eigenvalues of the Schr\"odinger operator $L=\Delta_g+q$ depending on integral quantities of the potential $q$ and a conformal invariant called the min-conformal volume. Moreover, when the Schr\"odinger…
In this paper, we extend the Hijazi inequality, involving the Energy-Momentum tensor, for the eigenvalues of the Dirac operator on $Spin^c$ manifolds without boundary. The limiting case is then studied and an example is given.
We are concerned with the dependence of the lowest positive eigenvalue of the Dirac operator on the geometry of rectangles, subject to infinite-mass boundary conditions. We conjecture that the square is a global minimiser both under the…
We investigate the second Dirac eigenvalue on Riemannian manifolds admitting a Killing spinor. In small dimensions the whole Dirac spectrum depends on special eigenvalues on functions and 1-forms. We compute and discuss the formulas in…
In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first $k$ eigenvalues which…
It is known that, for Dirac operators on Riemann surfaces twisted by line bundles with Hermitian-Einstein connections, it is possible to obtain estimates for the first eigenvalue in terms of the topology of the twisting bundle \cite{JL2}.…
We describe a new family of examples of hypersurfaces in the sphere satisfying the limiting-case in C. B\"ar's extrinsic upper bound for the smallest eigenvalue of the Dirac operator.
We estimate the behavior of the generalized eigenfunctions of critical Dirac operators (which are Dirac operators with eigenfunctions and/or resonances for $E=m$) plus small perturbations in the potential. The results also apply for other…
We derive an analytical expression for the distribution of the k-th smallest Dirac eigenvalue in QCD with imaginary isospin chemical potential in the Dirac operator. Because of its dependence on the pion decay constant F through the…
Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We study the…
We extend to the eigenvalues of the hypersurface Spin$^c$ Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Futhermore, we prove a correspondence between the existence of a Spin$^c$ Killing spinor…
In this largely expository paper we give a self-contained treatment of the Dirac operator. Emphasizing the algebraic point of view we first sketch the necessary prerequisites from Clifford algebras and their representations and then define…
Given a compact Riemannian spin manifold with positive scalar curvature, we find a family of connections $\nabla^{A_t}$ for $t\in[0,1]$ on a trivial vector bundle of sufficiently high rank, such that the first eigenvalue of the twisted…
We obtain sharp lower bounds for the first eigenvalue of four types of eigenvalue problem defined by the bi-Laplace operator on compact manifolds with boundary and determine all the eigenvalues and the corresponding eigenfunctions of a…
When an eigenvector of a semi-bounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-like…
In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general…
New exact upper and lower bounds are derived on the spectrum of the square of the hermitian Wilson Dirac operator. It is hoped that the derivations and the results will be of help in the search for ways to reduce the cost of simulations…