Spectral asymptotics of periodic elliptic operators

Ola Bratteli; Palle E.T. Jørgensen; Derek W. Robinson

doi:10.1007/PL00004773

Spectral asymptotics of periodic elliptic operators

Ola Bratteli^*, Palle E.T. Jørgensen, Derek W. Robinson

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

We demonstrate that the structure of complex second-order strongly elliptic operators H on R^d with coefficients invariant under translation by Z^d can be analyzed through decomposition in terms of versions H_z, z ∈ T^d, of H with z-periodic boundary conditions acting on L₂(I^d) where I = [0,1〉. If the semigroup S generated by H has a Hõlder continuous integral kernel satisfying Gaussian bounds then the semigroups S^z generated by the H_z have kernels with similar properties and z → S^z extends to a function on C^d\{0} which is analytic with respect to the trace norm. The sequence of semigroups S^(m),z obtained by rescaling the coefficients of H_z by c(x) → c(mx) converges in trace norm to the semigroup Ŝ^z generated by the homogenization Ĥ_z of H_z. These convergence properties allow asymptotic analysis of the spectrum of H.

Original language	English
Pages (from-to)	621-650
Number of pages	30
Journal	Mathematische Zeitschrift
Volume	232
Issue number	4
DOIs	https://doi.org/10.1007/PL00004773
Publication status	Published - 1999

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title = "Spectral asymptotics of periodic elliptic operators",

abstract = "We demonstrate that the structure of complex second-order strongly elliptic operators H on Rd with coefficients invariant under translation by Zd can be analyzed through decomposition in terms of versions Hz, z ∈ Td, of H with z-periodic boundary conditions acting on L2(Id) where I = [0,1〉. If the semigroup S generated by H has a H{\~o}lder continuous integral kernel satisfying Gaussian bounds then the semigroups Sz generated by the Hz have kernels with similar properties and z → Sz extends to a function on Cd\textbackslash{}\{0\} which is analytic with respect to the trace norm. The sequence of semigroups S(m),z obtained by rescaling the coefficients of Hz by c(x) → c(mx) converges in trace norm to the semigroup Ŝz generated by the homogenization Ĥz of Hz. These convergence properties allow asymptotic analysis of the spectrum of H.",

author = "Ola Bratteli and J{\o}rgensen, \{Palle E.T.\} and Robinson, \{Derek W.\}",

year = "1999",

doi = "10.1007/PL00004773",

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T1 - Spectral asymptotics of periodic elliptic operators

AU - Bratteli, Ola

AU - Jørgensen, Palle E.T.

AU - Robinson, Derek W.

PY - 1999

Y1 - 1999

N2 - We demonstrate that the structure of complex second-order strongly elliptic operators H on Rd with coefficients invariant under translation by Zd can be analyzed through decomposition in terms of versions Hz, z ∈ Td, of H with z-periodic boundary conditions acting on L2(Id) where I = [0,1〉. If the semigroup S generated by H has a Hõlder continuous integral kernel satisfying Gaussian bounds then the semigroups Sz generated by the Hz have kernels with similar properties and z → Sz extends to a function on Cd\{0} which is analytic with respect to the trace norm. The sequence of semigroups S(m),z obtained by rescaling the coefficients of Hz by c(x) → c(mx) converges in trace norm to the semigroup Ŝz generated by the homogenization Ĥz of Hz. These convergence properties allow asymptotic analysis of the spectrum of H.

AB - We demonstrate that the structure of complex second-order strongly elliptic operators H on Rd with coefficients invariant under translation by Zd can be analyzed through decomposition in terms of versions Hz, z ∈ Td, of H with z-periodic boundary conditions acting on L2(Id) where I = [0,1〉. If the semigroup S generated by H has a Hõlder continuous integral kernel satisfying Gaussian bounds then the semigroups Sz generated by the Hz have kernels with similar properties and z → Sz extends to a function on Cd\{0} which is analytic with respect to the trace norm. The sequence of semigroups S(m),z obtained by rescaling the coefficients of Hz by c(x) → c(mx) converges in trace norm to the semigroup Ŝz generated by the homogenization Ĥz of Hz. These convergence properties allow asymptotic analysis of the spectrum of H.

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U2 - 10.1007/PL00004773

DO - 10.1007/PL00004773

M3 - Article

SN - 0025-5874

VL - 232

SP - 621

EP - 650

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 4

ER -

Spectral asymptotics of periodic elliptic operators

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