Pseudospectra of differential operators

We study the pseudospectral theory of a variety of non-self- adjoint constant coecient and variable coecient dierential operators, showing that the phenomenon of non-trivial pseudospectra is typical rather than exceptional. We prove that the pseudospectra provide more stable in- formation about the operators under various limiting procedures than does the spectrum.

[1]  B. Simon,et al.  Scattering theory for systems with different spatial asymptotics on the left and right , 1978 .

[2]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[3]  A. Böttcher.  Infinite matrices and projection methods , 1995 .

[4]  The twisting trick for double well Hamiltonians , 1982 .

[5]  Lloyd N. Trefethen,et al.  Pseudospectra of Linear Operators , 1997, SIAM Rev..

[6]  H. J. Landau.  The notion of approximate eigenvalues applied to an integral equation of laser theory , 1977 .

[7]  H. J. Landau,et al.  Loss in unstable resonators , 1976 .

[8]  V. Liskevich.  On $C_0$-semigroups generated by elliptic second order differential expressions on $L^p$-spaces , 1996, Differential and Integral Equations.

[9]  S. C. Reddy.  Pseudospectra of Wiener-Hopf Integral Operators and Constant-Coefficient Differential Operators , 1993 .

[10]  A. Böttcher.  Pseudospectra and Singular Values of Large Convolution Operators , 1994 .

[11]  H. Landau,et al.  On Szegö’s eingenvalue distribution theorem and non-Hermitian kernels , 1975 .

[12]  Michael Reed,et al.  Methods of modern mathematical physics (vol.) I : functional analysis / Reed Michael, Barry Simon , 1980 .

[13]  L. Trefethen,et al.  Eigenvalues and pseudo-eigenvalues of Toeplitz matrices , 1992 .

[14]  M. G. Krein,et al.  Integral equations on a half-line with kernels de-pending upon the difference of the arguments , 1962 .

[15]  Lloyd N. Trefethen,et al.  Pseudospectra of the Convection-Diffusion Operator , 1993, SIAM J. Appl. Math..

[16]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[17]  Kim-Chuan Toh,et al.  Calculation of Pseudospectra by the Arnoldi Iteration , 1996, SIAM J. Sci. Comput..

[18]  E. Davies,et al.  One-parameter semigroups , 1980 .