Global Propagator for the Massless Dirac Operator and Spectral Asymptotics

Spectral theoretic characterization of the massless Dirac operator

We consider an elliptic self-adjoint first-order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the...

Inverse Problem for Interior Spectral Data of the Dirac Operator with Discontinuous Conditions

In this paper, we study the inverse problem for Dirac differential operators with  discontinuity conditions in a compact interval. It is shown that the potential functions can be uniquely determined by the value of the potential on some interval and parts of two sets of eigenvalues. Also, it is shown that the potential function can be uniquely determined by a part of a set of values of eigenfun...

Spectral analysis of the massless Dirac operator on a 3-manifold

The talk will give an overview of our further development of the paper [7] by Robert Downes, Michael Levitin and Dmitri Vassiliev and it will also give an insight how spectrum of massless Dirac operator on a 3-manifold interplays with geometric contents of the manifold. In contrast to the Riemann flat manifold studied in [7], 3-torus, we study the massless Dirac operator on a 3-sphere equipped ...

Spectral asymmetry of the massless Dirac operator on a 3 - torus

Root Asymptotics of Spectral Polynomials for the Lamé Operator

The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap pote...