Topological obstructions to the diagonalisation of pseudodifferential systems

Abstract

Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur.

v3: final version

References in corpus (4)

• Invariant subspaces of elliptic systems II: spectral theory
• Invariant subspaces of elliptic systems I: pseudodifferential projections
• Diagonalization of elliptic systems via pseudodifferential projections
• Spectral properties of the Neumann-Poincaré operator in 3D elasticity