Wave packet pseudomodes of twisted Toeplitz matrices

Communications on Pure and Applied Mathematics

Volumn 57, Issue 9, 2004, Pages 1233-1264

  Trefethen, Lloyd N ^a   Chapman, S J ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4043082073     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.20034     Document Type: Article

References (49)

1. Beam, R. W.; Warming, R. F. The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices. SIAM J. Sci. Comput. 14 (1993), no. 4, 971-1006.
2. Bender, C. M.; Boettcher, S.; Meisinger, P. N. PT-symmetric quantum mechanics. J. Math. Phys. 40 (1999), no. 5, 2201-2229.
3. Bender, C. M.; Brody, D. C.; Jones, H. F. Complex extension of quantum mechanics. Phys. Rev. Lett. 89 (2002), no. 27, 270401-1-270401-4.
4. Bender, C. M.; Orszag, S. A. Advanced mathematical methods for scientists and engineers. McGraw-Hill, New York, 1978.
5. Borthwick, D.; Uribe, A. On the pseudospectra of Berezin-Toeplitz operators. Methods Appl. Anal., to appear.
6. Böttcher, A. Pseudospectra and singular values of large convolution operators. J. Integral Equations Appl. 6 (1994), no. 3, 267-301.
7. Böttcher, A.; Embree, M.; Sokolov, V. I. The spectra of large Toeplitz band matrices with a randomly perturbed entry. Math. Comp. 72 (2003), no. 243, 1329-1348.
8. Böttcher, A.; Silbermann, B. Introduction to large truncated Toeplitz matrices. Universitext. Springer, New York, 1999.
9. Bube, K. P.; Strikwerda, J. C. Interior regularity estimates for elliptic systems of difference equations. SIAM J. Numer. Anal. 20 (1980), no. 4, 653-670.
10. Chapman, S. J. Subcritical transition in channel flows. J. Fluid Mech. 451 (2002), 35-97.
11. Davies, E. B. Pseudo-spectra, the harmonic oscillator and complex resonances. Proc. Roy. Soc. Lond. A 455 (1999), 585-599.
12. Davies, E. B. Semi-classical states for non-self-adjoint Schrödinger operators. Comm. Math. Phys. 200 (1999), 585-599.
13. Davies, E. B. Pseudospectra of differential operators. J. Operator Theory 43 (2000), 243-262.
14. Dencker, N.; Sjöstrand, J.; Zworski, M. Pseudospectra of semi-classical (pseudo)differential operators. Comm. Pure Appl. Math., to appear.
15. Diaconis, P. The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93 (1996), 1659-1664.
16. Diaconis, P.; Graham, R. L.; Morrison, J. A. Asymptotic analysis of a random walk on a hypercube with many dimensions. Random Structures Algorithms 1 (1990), 51-72.
17. Duistermaat, J. J.; Sjöstrand, J. A global construction for pseudo-differential operators with non-involutive characteristics. Invent. Math. 20 (1973), 209-225.
18. Embree, M.; Trefethen, L. N. Pseudospectra gateway. Web site located at: http://web.comlab.ox.ac.uk/projects/pseudospectra/
19. Gohberg, I. C.; Fel'dman, I. A. Convolution equations and projection methods for their solution. Translations of Mathematical Monographs, 41. American Mathematical Society, Providence, R.I., 1974.
20. Hatano, N.; Nelson, D. R. Vortex pinning and non-Hermitian quantum mechanics. Phys. Rev. B 56 (1997), 8651-8673.
21. Higham, N. J.; Knight, P. A. Matrix powers in finite precision arithmetic. SIAM J. Matrix Anal. Appl. 16 (1995), 343-358.
22. Hörmander, L. Differential equations without solutions. Math. Ann. 140 (1960), 169-173.
23. Hörmander, L. On the existence and the regularity of solutions of linear pseudo-differential equations. Enseignement Math. (2) 17 (1971), 99-163.
24. Hörmander, L. The analysis of linear partial differential operators. IV. Fourier integral operators. Grundlehren der Mathematischen Wissenschaften, 275. Springer, Berlin, 1985.
25. Irwin, M. C. On the stable manifold theorem. Bull. Lond. Math. Soc. 2 (1970), 196-198.
26. Jónsson, G. F.; Trefethen, L. N. A numerical analyst looks at the "cutoff phenomenon" in card shuffling and other Markov chains. Numerical analysis 1997 (Dundee), 150-178. Pitman Research Notes in Mathematics Series, 380. Longman, Harlow, 1998.
27. Kac, M.; Murdock, W. L.; Szegö, G. On the eigenvalues of certain Hermitian forms. J. Rational Mech. Anal. 2 (1953), 767-800.
28. Landau, H. J. On Szegö's eigenvalue distribution theory and non-Hermitian kernels. J. Analyse Math. 28 (1975), 335-357.
29. Landau, H. J. The notion of approximate eigenvalues applied to an integral equation of laser theory. Quart. Appl. Math. 35 (1977), 165-172.
30. Lewy, H. An example of a smooth linear partial differential equation without solution. Ann. Math. 66 (1957), 155-158.
31. Majda, A.; McDonough, J.; Osher, S. The Fourier method for nonsmooth initial data. Math. Comp. 32 (1978), 1041-1081.
32. Reddy, S. C. Pseudospectra of Wiener-Hopf integral operators and constant-coefficient differential operators. J. Integral Equations Appl. 5 (1993), 369-403.
33. Reddy, S. C.; Schmid, P. J.; Henningson, D. S. Pseudospectra of the Orr-Sommerfeld operator. SIAM J. Appl. Math. 53 (1993), 15-47.
34. Reichel, L.; Trefethen, L. N. Eigenvalues and pseudo-eigenvalues of Toeplitz matrices. Directions in matrix theory (Auburn, AL, 1990). Linear Algebra Appl. 162/164 (1992), 153-185.
35. Shub, M. Global stability of dynamical systems. Springer, New York, 1987.
36. Stevenson, R. P. On the validity of local mode analysis of multi-grid methods. Doctoral dissertation, University of Utrecht, 1990.
37. Stewart, G. W. On the powers of a matrix with perturbations. University of Maryland College Park Institute for Advanced Computer Studies TR-2001-91, Department of Computer Science TR-4317, revised 2002.
38. Taylor, M. E. Pseudodifferential operators. Princeton Univ. Press, Princeton, N.J., 1981.
39. Tilli, P. Locally Toeplitz sequences: spectral properties and applications. Linear Algebra Appl. 278 (1998), no. 1-3, 91-120.
40. Trefethen, L. N. Pseudospectra of matrices. Numerical analysis 1991 (Dundee, 1991), 234-266. Pitman Research Notes in Mathematics Series, 260. Longman, Harlow, 1992.
41. Trefethen, L. N. Pseudospectra of linear operators. SIAM Rev. 39 (1997), no. 3, 383-406.
42. Trefethen, L. N. Computation of pseudospectra. Acta numerica, 1999, 247-295. Acta Numer., 8. Cambridge Univ. Press, Cambridge, 1999.
43. Tyrtyshnikov, E. E. A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl. 232 (1996), 1-43.
44. Vaillancourt, R. On the stability of Friedrichs' scheme and the modified Lax-Wendroff scheme. Math. Comp. 24 (1970), 767-770.
45. Wilkinson, J. H. The algebraic eigenvalue problem. Clarendon, Oxford, 1965.
46. Wright, T. G. EigTool: a graphical tool for nonsymmetric eigenproblems. Software package available at: http://web.comlab.ox.ac.uk/projects/ pseudospectra/eigtool/
47. Wright, T. G. Algorithms and software for pseudospectra. Ph.D. thesis, Oxford University Computing Laboratory, 2002.
48. Zworski, M. A remark on a paper of E. B Davies. Proc. Amer. Math. Soc. 129 (2001), no. 10, 2955-2957.
49. Zworski, M. Numerical linear algebra and solvability of partial differential equations. Comm. Math. Phys. 229 (2002), no. 2, 293-307.