Invariant subspaces of elliptic systems I: Pseudodifferential projections

Capoferri, Matteo; Vassiliev, Dmitri

doi:10.1016/j.jfa.2022.109402

Cited by 11 publications

(9 citation statements)

References 41 publications

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“…In this paper we continue the analysis of invariant subspaces of elliptic systems initiated in [12], focussing on the spectral-theoretic aspects of the problem.…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…As in [12], we denote by C 1 .M / the linear space of m-columns of smooth complex-valued half-densities over M and by L 2 .M / its closure with respect to the inner product hv; wi WD Z M v w dx;…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…Invariant subspaces of elliptic systems II: Spectral theory 303 (b) [12,Theorem 2.5] We have P j AP j 0 mod ‰ 1 for j D 1; : : : ; m C ; P j AP j 0 mod ‰ 1 for j D 1; : : : ; m : Theorem 1.2 tells us that, given an elliptic self-adjoint operator A 2 ‰ s , one can construct a unique orthonormal basis of pseudodifferential projections commuting with A. These projections partition L 2 .M / into m invariant subspaces under the action of A, modulo C 1 .M /.…”

Section: Statement Of the Problemmentioning

confidence: 99%

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Invariant subspaces of elliptic systems II: Spectral theory

Capoferri

Vassiliev

2022

J. Spectr. Theory

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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. We show that the spectrum of A decomposes, up to an error with superpolynomial decay, into m distinct series, each associated with one of the eigenvalues of the principal symbol of A. These spectral results are then applied to the study of propagation of singularities in hyperbolic systems. The key technical ingredient is the use of the carefully devised pseudodifferential projections introduced in the first part of this work, which decompose L 2 .M / into almost-orthogonal almost-invariant subspaces under the action of both A and the hyperbolic evolution.

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“…In this paper we continue the analysis of invariant subspaces of elliptic systems initiated in [12], focussing on the spectral-theoretic aspects of the problem.…”

Section: Statement Of the Problemmentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

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Invariant subspaces of elliptic systems II: Spectral theory

Capoferri

Vassiliev

2022

J. Spectr. Theory

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“…However, diagonalising a system may not be possible due to topological obstructions, as is the case for some important physically meaningful operators; see Section 3. For this reason, other approaches to study the spectrum of systems, such as the use of pseudodifferential projections [7,8], are perhaps more natural, in that they always work and circumvent topological obstructions altogether.…”

Section: Statement Of the Problemmentioning

confidence: 99%

Topological obstructions to the diagonalisation of pseudodifferential systems

Capoferri¹,

Rozenblum²,

Saveliev³

et al. 2022

Preprint

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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.

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“…The principal symbol of this operator can be chosen as the matrix composed of the column vectors e ι (x, ξ), ι = 1, 2, 3. The procedure in [9], see also [11] describes the procedure of the construction of the consequent terms in the symbol of T, we, however, do not need them.…”

Section: Eigenvalue Asymptoticsmentioning

confidence: 99%

Discrete spectrum of zero order pseudodifferential operators

Rozenblum¹

2021

Preprint

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We study the rate of convergence of eigenvalues to the endpoints of essential spectrum for zero order pseudodifferential operators on a compact manifold.where µ µ µ ι (x, ω) are eigenvalues of the symbol a 0 (x, ω) (this simple but important fact was established in [1] in the scalar case and in [24] in the vector case). Recently, some applications required an analysis of the essential spectrum of such operators, see [14], [12], [15].

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Invariant subspaces of elliptic systems I: Pseudodifferential projections

Cited by 11 publications

References 41 publications

Invariant subspaces of elliptic systems II: Spectral theory

Invariant subspaces of elliptic systems II: Spectral theory

Topological obstructions to the diagonalisation of pseudodifferential systems

Discrete spectrum of zero order pseudodifferential operators

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