Journal article
Spectral asymptotics of periodic elliptic operators
Mathematische Zeitschrift, Vol.232(4), pp.621-650
12/1999
DOI: 10.1007/PL00004773
Abstract
We demonstrate that the structure of complex second-order strongly elliptic operators H on ${\bf R}^d$ with coefficients invariant under translation by ${\bf Z}^d$ can be analyzed through decomposition in terms of versions $H_z$ , $z\in{\bf T}^d$ , of H with z-periodic boundary conditions acting on $L_2({\bf I}^d)$ where ${\bf I}=[0,1\rangle$ . If the s emigroup S generated by H has a Hölder continuous integral kernel satisfying Gaussian bounds then the semigroups $S^z$ generated by the $H_z$ have kernels with similar properties and $z\mapsto S^z$ extends to a function on ${\bf C}^d\backslash\{0\}$ 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 $c(x)\to c(mx)$ converges in trace norm to the semigroup ${\widehat S}^z$ generated by the homogenization ${\widehat H}_z$ of $H_z$ . These convergence properties allow asymptotic analysis of the spectrum of H.
Details
- Title: Subtitle
- Spectral asymptotics of periodic elliptic operators
- Creators
- Ola Bratteli - Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia (e-mail: Derek.Robinson@anu.edu.au) AUPalle E.T Jørgensen - Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia (e-mail: Derek.Robinson@anu.edu.au) AUDerek W Robinson - Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia (e-mail: Derek.Robinson@anu.edu.au) AU
- Resource Type
- Journal article
- Publication Details
- Mathematische Zeitschrift, Vol.232(4), pp.621-650
- DOI
- 10.1007/PL00004773
- ISSN
- 0025-5874
- eISSN
- 1432-1823
- Publisher
- Springer-Verlag; Berlin Heidelberg
- Language
- English
- Date published
- 12/1999
- Academic Unit
- Mathematics
- Record Identifier
- 9983985805902771
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