Diagonalization of elliptic systems via pseudodifferential projections

Capoferri, Matteo

doi:10.1016/j.jde.2021.12.032

Cited by 10 publications

(13 citation statements)

References 49 publications

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“…We will need this in the special case of operators on the Euclidean space (thus certain topological obstacles arising in a more general case disappear.) We use the constructions elaborated in [9], [10].…”

Section: Extraction Of the Scalar Operatormentioning

confidence: 99%

“…Assuming that Condition A or Condition B is satisfied, q * E admits a global section v(x, ξ). Having such global section, the construction in [9] produces an isometric pseudodifferential operator T from L 2 (R d ) onto H 1 such that T * A 1 T is, up to an infinitely smoothing operator, a scalar pseudodifferential pseudodifferential operator. We, in fact, do not need an infinitely smoothing error, it is sufficient to have an error operator of order −2.…”

Section: Extraction Of the Scalar Operatormentioning

confidence: 99%

“…Note that in [9], a procedure of finding the required transformation is presented in a technically different way, which gives an explicit expression for the lower order symbol of A 1 not using contour integration.…”

Section: Extraction Of the Scalar Operatormentioning

confidence: 99%

“…Following our general approach, we consider the operator 1 − K, for which the principal symbol 1 1 1 1 1 1 1 1 1 − k, where 1 1 1 1 1 1 1 1 1 is the 3 × 3 unit matrix. The eigenvector e 1 (x, ξ) corresponding to the eigenvalue µ µ µ 1 (x) equals are smooth local eigenvector branches corresponding to the eigenvalues µ µ µ 2 (x, ξ), µ µ µ 3 (x, ξ) By the construction in [9], there is a unitary operator T transforming A to a diagonal form. 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.…”

Section: Eigenvalue Asymptoticsmentioning

confidence: 99%

“…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%

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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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Section: Extraction Of the Scalar Operatormentioning

confidence: 99%

Section: Extraction Of the Scalar Operatormentioning

confidence: 99%

Section: Extraction Of the Scalar Operatormentioning

confidence: 99%

Section: Eigenvalue Asymptoticsmentioning

confidence: 99%

Section: Eigenvalue Asymptoticsmentioning

confidence: 99%

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Discrete spectrum of zero order pseudodifferential operators

Rozenblum¹

2021

Preprint

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Global Propagator for the Massless Dirac Operator and Spectral Asymptotics

Capoferri

Vassiliev

2022

Integr. Equ. Oper. Theory

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We construct the propagator of the massless Dirac operator W on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals—the positive and the negative propagators—correspond to positive and negative eigenvalues of W, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of W.

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The discrete spectrum of the Neumann-Poincaré operator in 3D elasticity

Rozenblum

2023

J. Pseudo-Differ. Oper. Appl.

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For the Neumann-Poincaré (double layer potential) operator in the three-dimensional elasticity we establish asymptotic formulas for eigenvalues converging to the points of the essential spectrum and discuss geometric and mechanical meaning of coefficients in these formulas. In particular, we establish that for any body, there are infinitely many eigenvalues converging from above to each point of the essential spectrum. On the other hand, if there is a point where the boundary is concave (in particular, if the body contains cavities) then for each point of the essential spectrum there exists a sequence of eigenvalues converging to this point from below. The reasoning is based upon the representation of the Neumann-Poincaré operator as a zero order pseudodifferential operator on the boundary and the earlier results by the author on the eigenvalue asymptotics for polynomially compact pseudodifferential operators.

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Diagonalization of elliptic systems via pseudodifferential projections

Cited by 10 publications

References 49 publications

Discrete spectrum of zero order pseudodifferential operators

Discrete spectrum of zero order pseudodifferential operators

Global Propagator for the Massless Dirac Operator and Spectral Asymptotics

The discrete spectrum of the Neumann-Poincaré operator in 3D elasticity

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