Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-28T16:16:16.256Z Has data issue: false hasContentIssue false
  • English
  • Français

An Indefinite Convection-Diffusion Operator

Published online by Cambridge University Press:  01 February 2010

E.B. Davies
E.B. Davies
Affiliation:
Department of Mathematics, King's College, Strand, London WC2R 2LS, United Kingdom, E.Brian.Davies@kcl.ac.uk, http://www.mth.kcl.ac.uk/staff/eb_davies.html

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a mathematically rigorous analysis which confirms the surprising results in a recent paper of Benilov, O‘Brien and Sazonov [J. Fluid Mech. 497 (2003) 201-224] about the spectrum of a highly singular non-self-adjoint operator that arises in a problem in fluid mechanics. We also show that the set of eigenvectors does not form a basis for the operator.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 288 - 306
Copyright
Copyright © London Mathematical Society 2007

References

1.Bender, Carl M. and Boettcher, Stefan, ‘Real spectra in non-Hermitian Hamiltonians having PT symmetry’, Phys. Rev. Lett. 80 (1998) 5243–5246.CrossRefGoogle Scholar
2.Bender, Carl M., Brody, Dorje C., and Jones, Hugh F., ‘Complex extension of quantum mechanics’, Phys. Rev. Lett. 89, No.27 (30 Dec. 2002) 270401, 1–4.CrossRefGoogle ScholarPubMed
3.Benilov, E. S., ‘Explosive instability in a linear system with neutrally stable eigenmodes, part 2, multi-dimensional disturbances’, J. Fluid Mech. 501 (2004) 105–124.CrossRefGoogle Scholar
4.Benilov, E. S., O'Brien, S.B.G. and Sazonov, I. A., ‘A new type of instability: explosive disturbances in a liquid film inside a rotating horizontal cylinder’, J. Fluid Mech. 497 (2003) 201–224.CrossRefGoogle Scholar
5.Böttcher, A. and Silbermann, B., Introduction to large truncated Toeplitz matrices (Springer, New York, 1999).CrossRefGoogle Scholar
6.Chugunova, M. and Pelinovsky, D., ‘Spectrum of an non-self-adjoint operator associated with the cylindric heat equation’, preprint, Feb. 2007.Google Scholar
7.Davies, E. B., ‘Pseudospectra, the harmonic oscillator and complex resonances’, Proc. Roy. Soc. London A 455 (1999) 585–599.CrossRefGoogle Scholar
8.Davies, E. B., ‘Semi-classical states for non-self-adjoint Schrödinger operators’, Commun. Math. Phys. 200 (1999) 35’41.CrossRefGoogle Scholar
9.Davies, E. B., ‘Wild spectral behaviour of anharmonic oscillators’, Bull. London Math. Soc. 32 (2000) 432–438.CrossRefGoogle Scholar
10.Davies, E. B., Linear operators and their spectra, Cambridge Studies in Advanced Math. 106 (Cambridge Univ. Press, 2007).CrossRefGoogle Scholar
11.Davis, E. B. and Kuijlaars, A., ‘Spectral asymptotics of the non-self- adjoint harmonic oscillator’, J. London Math. Soc. (2) 70 (2004) 420–426.CrossRefGoogle Scholar
12.Dorey, P., Dunning, C. and Tateo, R., ‘Differential equations and the Bethe ansatz’, J. Phys. A, Math. Gen. 34 (2001) 5679.Google Scholar
13.Dorey, P., Dunning, C. and Tateo, R., ‘The ODE/IM correspondence’, preprint, arXiv:hep-th/0703066, March 2007.CrossRefGoogle Scholar
14.Lenferink, H.W.J. and Spijker, M. N., ‘On the use of stability regions in the numerical analysis of initial value problems’, Math. Comp. 57 (1991) 221–237.CrossRefGoogle Scholar
15.Levy, H. and Lessman, F., Finite difference equations (Dover, New York 1992).Google Scholar
16.Shin, K. C., ‘Eigenvalues of PT-symmetric oscillators with polynomial potentials’, J. Phys.A 38 (2005) 6147–6166.Google Scholar
17.Trefethen, L. N.‘Pseudospectra of matrices’, Numerical analysis 1991, ed. Griffiths, D. F. and Watson, G. A., Pitman Res. Notes Math. Ser. 260(Longman Scientific and Technical, 1992) 234–266.Google Scholar
18.Trefethen, L. N. and Chapman, S. J., ‘Wave packet pseudomodes of twisted Toeplitz matrices’ Cornrn. Pure Appl. Math. 57 (2004) 1233–1264.CrossRefGoogle Scholar
19.Trefethen, L. N. and Embree, M., Spectra and pseudospectra (Princeton Univ. Press, 2005).CrossRefGoogle Scholar