TY - JOUR
T1 - Diagonalization of elliptic systems via pseudodifferential projections
AU - Capoferri, Matteo
N1 - Funding Information:
This work was supported by the Leverhulme Trust Research Project Grant RPG-2019-240 , which is gratefully acknowledged.
Funding Information:
I am grateful to Alex Strohmaier for raising questions that eventually led to this paper and for pointing out useful references; to Jean-Claude Cuenin for helpful comments on a preliminary version of this manuscript and for bibliographic suggestions; to Grigori Rozenbloum for insightful discussions on the role of topological obstructions; to Dmitri Vassiliev for numerous discussions on this and related topics throughout the years. This work was supported by the Leverhulme Trust Research Project Grant RPG-2019-240, which is gratefully acknowledged.
Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2022/3/15
Y1 - 2022/3/15
N2 - 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.
AB - 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.
KW - Elliptic systems
KW - Invariant subspaces
KW - Pseudodifferential projections
KW - Spectral asymptotics
KW - Unitary diagonalization
UR - http://www.scopus.com/inward/record.url?scp=85122523659&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2021.12.032
DO - 10.1016/j.jde.2021.12.032
M3 - Article
AN - SCOPUS:85122523659
SN - 0022-0396
VL - 313
SP - 157
EP - 187
JO - Journal of Differential Equations
JF - Journal of Differential Equations
ER -