E. B. Davies, Pseudospectra, the harmonic oscillator and complex resonances, Proc. Roy. Soc. Lond. A, 455 (1999), pp. 585-599.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....our results appears in the literature of microlocal analysis of di erential and pseudodi erential operators [41] In several papers beginning in 1996, E. B. Davies has shown that a Schr odinger operator with a complex potential may have wave packet pseudomodes for rapidly decreasing values of [13,14,15]. M. Zworski has pointed out that Davies example and its generalizations are instances of a general theory of variablecoe cient partial di erential and pseudodi erential operators developed by H ormander, Treves and others decades ago, originally motivated by the analysis of Lewy s phenomenon of ....

.... localized as in (13) or (45) it is not exponentially good as in (12) or (44) 10 Discussion As mentioned in the Introduction, the phenomena we have discussed for matrices have analogues for di erential and pseudodi erential operators, which have been investigated by Davies, Zworski, and others [6,13,14,15,16,52,53]. The analogy is close. For constant coecient nonselfadjoint di erential operators, one nds exponentially good pseudoeigenvectors localized at a boundary [15,35] just as for nonhermitian Toeplitz matrices, whereas when the coecients become variable, wave packet pseudoeigenvectors appear. In [16] ....

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E. B. Davies, Pseudo-spectra, the harmonic oscillator and complex resonances, Proc. Roy. Soc. Lond. A 455 (1999), 585-599.

.... matrices and operators were rst investigated in the 1970s and 1980s [21,35,55] and became a standard tool in the 1990s [8,48,49,51] with applications in uid mechanics [54] numerical analysis [29,42,46,47] operator theory [2,5] control theory [30] Markov chains [31] di erential equations [9,10,43], and integral equations [40,41] In all of these elds it has been found that in cases of pronounced nonnormality, eigenvalues and eigenvectors alone do not always reveal much about the aspects of the behavior of a matrix or operator that matter in applications, including phenomena of stability, ....

E. B. Davies, Pseudospectra, the harmonic oscillator and complex resonances, Proc. Roy. Soc. Lond. A, 455 (1999), pp. 585-599.

....the Breit Wigner peaks. Finer description is given in terms of resonances (see [20] for an introduction and pointers to references) and the relation between the two methods is going to be described elsewhere. A Simple Model Instead of considering a complicated system such as (5) we follow Davies [1] and study the rotated harmonic oscillator: 6) P = D 2 x e i x 2 ; 0 : The spectrum of this operator is easily computed by making a change of variables y = e i =4 x, so that P = e i =2 (D 2 y y 2 ) e i =2 P ; where P is the harmonic oscillator (1) With a ....

....we can see from this that the spectrum of P is given by e i =2 (2n 1) n = 0; 1; where spectrum is the set of z 2 C for which there exists u 2 L 2 (R) such that Pu = zu. We can now try to nd the same eigenvalues numerically. Although better accounts of this are available in [1] and [19, Sect.9] out of curiosity, I proceeded directly using Mathematica, c and a simple discretization based on taking as a basis of L 2 eigenfunctions, f j g 1 j=0 , of a di erent harmonic oscillator (D 2 x 2x 2 ) For a truncated basis of 200 elements, and the discretization ....

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E.B. Davies Pseudo-spectra, the harmonic oscillator and complex resonances. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455(1999), 585-599.

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E. B. Davies, Pseudospectra, the harmonic oscillator and complex resonances, Proc. Roy. Soc. Lond. A, 455 (1999), pp. 585-599.

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