Nonhermitian systems and pseudospectra
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Talk no. 10, 11 p.

Four applications are outlined of pseudospectra of highly nonnormal linear operators.

Trefethen, Lloyd N. 1

1 Oxford University Computing Laboratory, Wolfson Bldg., Parks Rd., Oxford OX1 3QD, UK
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Trefethen, Lloyd N. Nonhermitian systems and pseudospectra. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Talk no. 10, 11 p. http://archive.numdam.org/item/SEDP_2005-2006____A10_0/

[1] P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958), 1492–1505. 1958.

[2] C. M. Bender and S. Boettcher, Real spectra in non-Hermitian hamiltonians having 𝒫T-symmetry, Phys. Rev. Lett. 80 (1998), 5243–5246. | MR | Zbl

[3] E. S. Benilov, S. B. G. O’Brien, and I. A. Sazanov, A new type of instability: explosive disturbances in a liquid film inside a rotating horizontal cylinder, J. Fluid Mech. 497 (2003), 201–224. | Zbl

[4] L. Boberg and U. Brosa, Onset of turbulence in a pipe, Z. Naturforschung 43a (1988), 697–726.

[5] D. Borthwick and A. Uribe, On the pseudospectra of Berezin–Toeplitz operators, Methods Appl. Anal. 10 (2003), 31–65. | MR | Zbl

[6] Y.-J. Cheng, C. G. Fanning and A. E. Siegman, The experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes, Phys. Rev. Lett. 77 (1996), 627–630.

[7] C. Cossu and J. M. Chomaz, Global measures of local convective instabilities, Phys. Rev. Lett. 78 (1997), 4387–4390.

[8] E. B. Davies, Pseudo-spectra, the harmonic oscillator and complex resonances, Proc. Roy. Soc. A 455 (1999), 585–599. | MR | Zbl

[9] E. B. Davies, Sepctral properties of random non-self-adjoint matrices and operators, Proc. Roy. Soc. A 457 (2001), 191–206. | MR | Zbl

[10] E. B. Davies and A. B. J. Kuijlaars, Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. Lond. Math. Soc. 70 (2004), 420–426. | MR | Zbl

[11] N. Dencker, J. Sjöstrand, and M. Zworski, Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math. 57 (2004), 384–415. | MR | Zbl

[12] G. Domokos and P. Holmes, On nonlinear boundary value problems: ghosts, parasites, and discretizations, Proc. Roy. Soc. A 459 (2003), 1535–1561. | MR | Zbl

[13] M. Embree and L. N. Trefethen, Pseudospectra Gateway, http://www.comlab.ox.ac.uk/pseudospectra, 2001.

[14] A. G. Fox and T. Li, Resonant modes in a maser interferometer, Bell Syst. Tech. J. 40 (1961), 453–488.

[15] N. Hatano and D. R. Nelson, Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77 (1996), 570–573.

[16] N. Hatano and D. R. Nelson, Vortex pinning and non-Hermitian quantum mechanics, Phys. Rev. B 56 (1997), 8651–8673.

[17] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1980. | MR | Zbl

[18] H. J. Landau, The notion of approximate eigenvalues applied to an integral equation of laser theory, Quart. Appl. Math. 35 (1977), 165–172. | MR | Zbl

[19] H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. Math. 66 (1957), 155–158. | MR | Zbl

[20] K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Eq. 55 (1984), 225–256. | MR | Zbl

[21] K. Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding, IEEE J. Quant. Elect. QE-15 (1979), 566–570.

[22] A. L Schawlow and C. H. Townes, Infrared and optical masers, Phys. Rev. 112 (1958), 1940–1949.

[23] A. E. Siegman, Lasers, University Science Books, 1986.

[24] A. E. Trefethen, L. N. Trefethen and P. J. Schmid, Spectra and pseudospectra for pipe Poiseuille flow, Comp. Meth. Appl. Mech. Eng. 1926 (1999), 413–420. | MR | Zbl

[25] L. N. Trefethen, Wave packet pseudomodes of variable coefficient differential operators, Proc. Roy. Soc. A 461 (2005), 3099–3122. | MR

[26] L. N. Trefethen and S. J. Chapman, Wave packet pseudomodes of twisted Toeplitz matrices, Comm. Pure Appl. Math. 57 (2004), 1233–1264. | MR | Zbl

[27] L. N. Trefethen, M. Contedini and M. Embree, Spectra, pseudospectra and localization for random bidiagonal matrices, Comm. Pure Appl. Math. 54 (2001), 595–623. | MR | Zbl

[28] L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton U. Press, Princeton, 2005. | MR | Zbl

[29] L. N. Trefethen, A. E. Trefethen, S. C. Reddy and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261 (1993), 578–584. | MR

[30] T. G. Wright, EigTool package, http://www.comlab.ox.ac.uk/pseudospectra/eigtool, 2002.

[31] M. Zworski, A remark on a paper of E. B. Davies, Proc. Amer. Math. Soc. 129 (2001), 2955–2957. | MR | Zbl