Diagonalization of elliptic systems via pseudodifferential projections

TL;DR

This paper develops a method to approximately diagonalize elliptic self-adjoint pseudodifferential operators on closed manifolds using pseudodifferential projections, providing explicit formulas and spectral analysis insights.

Contribution

It introduces an invariant algorithm for computing the full symbol of an almost-unitary operator that diagonalizes such operators, advancing spectral analysis techniques.

Findings

Constructed an almost-unitary pseudodifferential operator that diagonalizes A modulo smoothing operators.

Provided explicit formulas for the full and subprincipal symbols of the diagonalizing operator.

Analyzed the relationship between the spectra of A and its diagonalization, with implications for spectral asymptotics.

Abstract

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential projections commuting with $A$, we construct an almost-unitary pseudodifferential operator that diagonalizes $A$ modulo an infinitely smoothing operator. We provide an invariant algorithm for the computation of its full symbol, as well as an explicit closed formula for its subprincipal symbol. Finally, we give a quantitative description of the relation between the spectrum of $A$ and the spectrum of its approximate diagonalization, and discuss the implications at the level of spectral asymptotics.