Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T12:19:31.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Bounds Using Higher-Order Numerical Ranges

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, [email protected], 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.

This paper describes how to obtain bounds on the spectrum of a non-self-adjoint operator by means of what are referred to here as ‘its higher-order numerical ranges’. Proofs of some of their basic properties are given, as well as an explanation of how to compute them. Finally, they are used to obtain new spectral insights into the non-self-adjoint Anderson model in one and two space dimensions.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 17 - 45
Copyright
Copyright © London Mathematical Society 2005

References

1.Abramov, A., Aslanyan, A. and Davies, E. B., ‘Bounds on complex eigenvalues and resonances’, J. Phys. A Math. Gen. 34 (2001)52–72.Google Scholar
2.Brown, B. M. and Eastham, M. S. P., ‘Analytic continuation and resonance free regions for Sturm—Liouville potentials with power decay’, J. Comp. Appl. Math. 148 (2002) 49–63.Google Scholar
3.Böttcher, A., ‘Pseudospectra and singular values of large convolution operators’, J. Integral Equations Appl. 6 (1994) 267–301.Google Scholar
4.Böttcher, A. and Silbermann, B., Introduction to large truncated Toeplitz matrices (Springer, New York, 1998).Google Scholar
5.Brézin, E. and Zee, A., ‘Non-Hermitean delocalization, multiple scattering and bounds’, Nuclear Phys. B509 (1998) 599–614.Google Scholar
6.Burke, J. and Greenbaum, A., ‘Some equivalent characterizations of the polynomial numerical hull of degree k’, Preprint, Nov. 2004; http://web.comlab.ox.ac.uk/oucl/publications/natr/na-04-29.html.Google Scholar
7.Dahmen, H. A., Nelson, D. R. and Shnerb, N. M., ‘Population dynamics and non-hermitian localization’, Statistical mechanics of biocomplexity, (ed. Reguera, D., Vilar, J. M. G. and Rubi, J. M., Springer, Berlin 1999) 124–151.Google Scholar
8.Davies, E. B., One-parameter semigroups (Academic Press, London, 1980).Google Scholar
9.Davies, E. B., ‘Spectral properties of random non-self-adjoint matrices and operators’, Proc. Roy. Soc. London A 457 (2001) 191–206.Google Scholar
10.Davies, E. B., ‘Spectral theory of pseudo-ergodic operators’, Commun. Math. Phys. 216 (2001) 687–704.Google Scholar
11.Davies, E. B. and Nath, J., ‘Schrödinger operators with slowly decaying potentials’, J. Comp. Appl. Math. 148 (2002) 1–28.Google Scholar
12.Davis, Ch. and Salemi, A., ‘On polynomial numerical hulls of normal matrices’, Lin. Alg. Appl. 383 (2004) 151–161.Google Scholar
13.Feinberg, J. and Zee, A., ‘Spectral curves of non-hermitian hamiltonians’, Nucl. Phys. B 552 (1999) 599–623.Google Scholar
14.Goldsheid, I. Ya. and Khoruzhenko, B. A., ‘Distribution of eigenvalues in non- Hermitian Anderson model’, Phys. Rev. Lett. 80 (1998) 2897–2901.Google Scholar
15.Goldsheid, I. Ya. and Khoruzhenko, B. A., ‘Eigenvalue curves of asymmetric tridiagonal random matrices’, Electronic J. Prob. 5 (2000) Paper 16, 1–28.Google Scholar
16.Goldsheid, I. Ya. and Khoruzhenko, B. A., ‘Regular spacings of complex eigenval ues in the one-dimensional non-Hermitian Anderson model’, Comm. Math. Phys. 238 (2003)505–524.Google Scholar
17.Greenbaum, A., ‘Generalizations of the field of values useful in the study of polynomial functions of a matrix’, Linear Algebra Appl. 347 (2002) 233–249.Google Scholar
18.Hatano, N. and Nelson, D. R., ‘Vortex pinning and non-Hermitian quantum mechanics’, Phys. Rev. B 56 (1997) 8651–8673.Google Scholar
19.Hatano, N. and Nelson, D. R., ‘Non-Hermitian delocalization and eigenfunctions’, Phys. Rev. B 58 (1998) 8384–8390.Google Scholar
20.Langer, H. and Tretter, C., ‘Spectral decomposition of some nonselfadjoint block operator matrices’, J. Operator Theory 39 (1998) 339–359.Google Scholar
21.Langer, H., Markus, A., Matsaev, V. and Tretter, C, ‘A new concept for block operator matrices. The quadratic numerical range’, Linear Algebra Appl. 330 (2001) 89–112.Google Scholar
22.Martínez, C., ‘The non-self-adjoint Anderson model’, J. Oper. Theory, to appear.Google Scholar
23.Nelson, D. R. and Shnerb, N. M., ‘Non-Hermitian localization and population biology’, Phys. Rev. E 58 (1998) 1383–1403.Google Scholar
24.Nevanlinna, O., Convergence of iterations for linear equations (Birkhauser, Basel, 1993).Google Scholar
25.Reddy, S. C., Schmid, P. J. and Henningson, D. S., ‘Pseudospectra of the Orr— Sommerfeld operator’, SIAM J. Appl. Math. 53 (1993) 15–47.Google Scholar
26.Redparth, P., ‘Spectral properties of non-self-adjoint operators in the semi-classical regime’, J. Differential Equations 177 (2001) 307–330.Google Scholar
27.Shkalikov, A., ‘The limit behaviour of the spectrum for large parameter values in a model problem’, Math. Notes 62 (1997) No. 6.Google Scholar
28.Stoller, S. D., Happer, W. and Dyson, F. J., ‘Transverse spin relaxation in inhomogeneous magnetic fields’, Phys. Rev. A 44 (1991) 7459–7477.Google Scholar
29.Trefethen, L. N., ‘Pseudospectra of matrices’, Numerical Analysis 1991 (ed. Griffiths, D. F. and Watson, G. A., Longman Sci. Tech. Publ., Harlow, UK, 1992) 234–266.Google Scholar
30.Trefethen, L. N. and Embree, M., Spectra and pseudospectra (Princeton Univ. Press, to appear).Google Scholar
31.Trefethen, L. N., Contedini, M. and Embree, M., ‘Spectra, pseudospectra, and localization for random bidiagonal matrices’, Comm. Pure Appl. Math. 54 (2001) 595–623.Google Scholar