## SPECTRAL INSTABILITY OF SEMICLASSICAL OPERATORS (2009)

### BibTeX

@MISC{Dencker09spectralinstability,

author = {Nils Dencker},

title = {SPECTRAL INSTABILITY OF SEMICLASSICAL OPERATORS},

year = {2009}

}

### OpenURL

### Abstract

We give a short review of the spectral instability of non-normal semiclassical differential operators, both for scalar operators and systems.

### Citations

391 | Introduction to the theory of linear nonself-adjoint operators - Gohberg, Krein - 1965 |

192 |
Spectral asymptotics in the semi-classical limit
- Dimassi, Sjöstrand
- 1999
(Show Context)
Citation Context ...tisfying 1 ≤ m(w) ≤ C(1 + |w − w0|) M m(w0) ∀ w, w0 ∈ T ∗ R n for some C and M. Then m is an admissible weight function and we can define the symbol classes P ∈ S(m) by ‖∂ α wP(w)‖ ≤ Cαm(w) Following =-=[23]-=- we can then define the semiclassical operator P(h) = P w (x,hD). In the analytic case we require that the symbol estimates hold in a tubular neighborhood of T ∗ R n : (3.10) ‖∂ α wP(w)‖ ≤ Cαm(Rew) fo... |

121 | Pseudospectra of linear operators
- Trefethen
- 1997
(Show Context)
Citation Context ...ense in the set of values for which the bracket is non-zero. In simple one dimensional examples we can already see that the spectrum, σ(P(h)), typically lies deep inside the pseudospectrum Λ(p) — see =-=[5, 6, 71]-=- for numerical examples of this phenomenon. Consider for example the following non-selfadjoint operator P(h) = (hDx) 2 + i(hDx) + x 2 . A formal conjugation e −x/2h P(h)e x/2h = (hDx) 2 + x 2 + 1 4 sh... |

96 | An introduction to semiclassical and microlocal analysis. Universitext - Martinez - 2002 |

91 | Perturbation theory for linear operators, Die Grundlehren der math - Kato - 1966 |

87 | Computation of pseudospectra
- Trefethen
- 1999
(Show Context)
Citation Context ... the accurate computation of eigenvalues of large non-normal matrices and has applications in a wide field, from random matrix theory, the stability of flows to things as mundane as brake squeal (see =-=[72]-=- for more examples). The standard example of the spectral instability of non-normal matrices is the following perturbation of the N × N Jordan matrix ⎛ ⎞ 0 1 0 ... 0 0 0 0 1 ... 0 0 ... . . . . . ⎜0 0... |

81 | Introduction to the Spectral Theory of Polynomial Operator - Markus - 1988 |

54 | Résonances en limite semi-classique - Helffer, Sjöstrand - 1986 |

50 | M.L.: On the Lidskii-Vishik-Lyusternik perturbation theory for the eigenvalues of matrices with arbitrary Jordan structure
- Moro, Burke, et al.
- 1997
(Show Context)
Citation Context ... = ε 1/N e iπ(1+2k)/N k = 1,...,N We find for large N that |λk| = ε 1/N ≈ 1, thus a small perturbation can give a large change of the eigenvalues. This holds in general, Lidskiǐ, in a pioneering work =-=[54, 58]-=-, showed that small perturbations A + εB of an N × N matrix A, could produce “Lidskii circles” of eigenvalues. Davies and Hager [16] has gone further, proving that for large N, most eigenvalues of ran... |

46 |
Pseudo-spectra of semi-classical (Pseudo)differential operators
- Dencker, Sjöstrand, et al.
- 2004
(Show Context)
Citation Context ...h (u) the semiclassical polarization set of u. We could similarly define the analytic semiclassical polarization set by using the FBI transform and analytic pseudodifferential operators, see (2.6) in =-=[22]-=-. Remark 3.16. The semiclassical polarization sets are closed, linear in the fiber and has the functorial properties of the C∞ polarization sets in [18]. In particular, we find that π(WF pol h (u) \ 0... |

45 |
Pseudospectra of the OrrSommerfeld operator
- Reddy, Schmid, et al.
- 1993
(Show Context)
Citation Context ...nsions of certain types of non-selfadjoint differential operators. The first to study the stability of the spectrum for non-selfadjoint differential operators seems to be Reddy, Schmid and Henningsen =-=[66]-=- who studied the complex Airy and the Orr-Sommerfeld equations. When studying the spectral instability of differential operators, it is illuminating (and physically relevant) to study semiclassical op... |

44 | Singularités analytiques microlocales - Sjöstrand - 1982 |

40 | An example of a smooth linear partial differential equation without solution - LEWY - 1957 |

28 | Regulating Pollution
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- 1998
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Citation Context ...ors could grow exponentially, see [1]. There also occurs “spectral pollution”, in the sense that the spectra of the finite dimensional projections do not converge to the spectrum of the operator, see =-=[14]-=- for a numerical example. Another important problem is the behaviour of evolution semigroups for non-normal operators and the relation to pseudospectrum, see for example [9]. In this paper, we shall r... |

28 | Elementary linear algebra for advanced spectral problems, Annales de l’Institut Fourier 57(2007 - Sjöstrand, Zworski |

25 |
On the propagation of polarization sets for systems of real principle type
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Citation Context ...ytic pseudodifferential operators, see (2.6) in [22]. Remark 3.16. The semiclassical polarization sets are closed, linear in the fiber and has the functorial properties of the C∞ polarization sets in =-=[18]-=-. In particular, we find that π(WF pol h (u) \ 0) = WFh(u) = ⋃ WFh(uj) if π is the projection along the fiber variables: π : T ∗Rn ×CN ↦→ T ∗Rn . We also find that } A(WF pol h (u)) = { (w,A(w)z) : (w... |

18 | Instabilité spectrale semi-classique d’opérateurs non-autoadjoints - Hager - 2005 |

18 | Determinants of pseudodifferential operators and complex deformations of phase space. Methods and Applications of Analysis - Melin, Sjöstrand |

15 | Spectral instability for some Schrödinger operators
- Aslanyan, Davies
(Show Context)
Citation Context ...perators is directly connected with the bracket condition and the solvability question. In these infinite dimensional problems, the condition numbers of the eigenvectors could grow exponentially, see =-=[1]-=-. There also occurs “spectral pollution”, in the sense that the spectra of the finite dimensional projections do not converge to the spectrum of the operator, see [14] for a numerical example. Another... |

15 |
Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators
- Hager, Sjostrand
(Show Context)
Citation Context ...ic non-normal quadratic Weyl operators [63]. Sjöstrand and Hager has proved that for certain random perturbations of pseudodifferential operators, the spectrum will satisfy a asymptotic Weyl law, see =-=[31]-=-. It is interesting to compare this to the recent proof by Tao and Vu [70] of the circular law for random matrices, for which the spectrum is uniformly distributed in a disk. 3. Systems of semiclassic... |

12 | Wave packet pseudomodes of twisted Toeplitz matrices
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Citation Context ...lassical properties of symbols and the existence of localized quasimodes can be also be observed in the Berezin-Toeplitz quantization of compact symplectic Kähler manifolds. See Chapman and Trefethen =-=[4]-=- for the case of the torus, and Borthwick and Uribe [2] for the general C ∞ case.INSTABILITY 9 For non-principal type scalar semiclassical operators, there are still many open questions about the pse... |

12 |
Differential operators of principal type
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- 1960
(Show Context)
Citation Context ...ies [6] for Schrödinger operators in one dimension, but as was pointed out by Zworski [74, 75],4 NILS DENCKER it follows in general from a simple adaptation of the now classical results of Hörmander =-=[34, 35]-=- and Duistermaat-Sjöstrand [25], and is connected with the solvability problem by the bracket condition. It follows from (2.3) that, unlike the case of normal operators, the resolvent blows up as any ... |

10 | On the pseudospectra of Berezin-Toeplitz operators, Methods and Applications of Analysis 10
- Borthwick, Uribe
- 2003
(Show Context)
Citation Context ...alized quasimodes can be also be observed in the Berezin-Toeplitz quantization of compact symplectic Kähler manifolds. See Chapman and Trefethen [4] for the case of the torus, and Borthwick and Uribe =-=[2]-=- for the general C ∞ case.INSTABILITY 9 For non-principal type scalar semiclassical operators, there are still many open questions about the pseudospectrum. Pravda-Starov has studied the pseudospectr... |

10 | Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra
- Boulton
(Show Context)
Citation Context ...1 ) k k+1 } ∩ Spec(P(h)) = ∅ 0 < h ≤ h1 We obtain (2.17) from [22, Theorem 1.4] and (2.17) from [68]. In one dimension, the resolvent estimate was proved in [73], and in some special cases by Boulton =-=[3]-=- who also showed that the bounds are optimal. As was demonstrated by Trefethen [71] this is also easy to see numerically. We have the following simple higher dimensional example from [22] to which we ... |

8 |
A global construction for pseudo-differential operators with non-involutive characteristics
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(Show Context)
Citation Context ...n one dimension, but as was pointed out by Zworski [74, 75],4 NILS DENCKER it follows in general from a simple adaptation of the now classical results of Hörmander [34, 35] and Duistermaat-Sjöstrand =-=[25]-=-, and is connected with the solvability problem by the bracket condition. It follows from (2.3) that, unlike the case of normal operators, the resolvent blows up as any power of h: ‖(P(h) − z) −1 ‖ ≥ ... |

8 |
Spectral pollution and second-order relative spectra for self-adjoint operators
- Levitin, Shargorodsky
(Show Context)
Citation Context ... the pseudospectral method in numerical analysis. There are several other ways of measuring spectral stability, for example the structured ε-pseudospectrum [24] and the second-order relative spectrum =-=[41]-=-. In the present review article, it will not be possible to make a more thorough treatment of the spectral instability of matrices. For more results, examples and references we refer the reader to [72... |

8 |
A general result about pseudo-spectrum for Schrödinger operators
- Pravda-Starov
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(Show Context)
Citation Context ... many points Λ(p) \ Λ−(p) quasimodes can exist, since that the vanishing of the Poisson bracket {Re p, Im p} is not enough to guarantee the absence of a quasimode. In fact, as proved by Pravda-Starov =-=[59]-=-, a violation of the condition (Ψ) (see [37, Section 26.4]) can produce quasimodes. Condition (Ψ) is directly connected to the Nirenberg-Treves conjecture, which says that a pseudodifferential operato... |

6 | Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle
- Davies, Simon
(Show Context)
Citation Context ... be a much larger set, as seen above for the Jordan matrices. Observe that when |z| ≥ ‖A‖ we have for any matrix A the estimate ‖(z IdN −A) −1 ( π ) ‖ ≤ cot dist(z, Spec(A)) 4N −1 by Davies and Simon =-=[17]-=-. The use of the resolvent norm has given rise to the pseudospectral method in numerical analysis. There are several other ways of measuring spectral stability, for example the structured ε-pseudospec... |

6 |
An optimal bound for the spectral variation of two matrices, ibid
- Elsner
- 1985
(Show Context)
Citation Context ...ince in general one has that the minimal distance (under permutations) between the spectra of two N × N matrices is bounded by (‖A‖ + ‖B‖) 1−1/N ‖A − B‖ 1/N where ‖A‖ is the standard matrix norm, see =-=[26]-=-. This spectral instability complicates the mathematical modelling of non-symmetric problems, since more accurate models usually Date: October 1, 2009. 2000 Mathematics Subject Classification. 35S05 (... |

5 | Perturbations of Jordan matrices
- Davies, Hager
(Show Context)
Citation Context ...genvalues. This holds in general, Lidskiǐ, in a pioneering work [54, 58], showed that small perturbations A + εB of an N × N matrix A, could produce “Lidskii circles” of eigenvalues. Davies and Hager =-=[16]-=- has gone further, proving that for large N, most eigenvalues of random perturbations of the N × N Jordan matrix will be very close to the unit circle. The spectral instability of the Jordan matrices ... |

5 | Micro-hyperbolic pseudo-differential operators - Kashiwara, Kawai - 1975 |

5 | On dissipation-induced destabilization and brake squeal: a perspective using structured pseudospectra - Kessler, O’Reilly, et al. - 2007 |

4 | Kuijlaars A B J: Spectral Asymptotics of the non-self-adjoint harmonic oscillator - Davies |

3 |
equations without solutions
- Differential
- 1960
(Show Context)
Citation Context ...ies [6] for Schrödinger operators in one dimension, but as was pointed out by Zworski [74, 75],4 NILS DENCKER it follows in general from a simple adaptation of the now classical results of Hörmander =-=[34, 35]-=- and Duistermaat-Sjöstrand [25], and is connected with the solvability problem by the bracket condition. It follows from (2.3) that, unlike the case of normal operators, the resolvent blows up as any ... |

2 |
Davies Pseudospectra, the harmonic oscillator and complex resonances
- B
- 1999
(Show Context)
Citation Context ...r, the Péclet number orINSTABILITY 3 the frequency. Then, the spectral instability can be defined as a function of h, see Definition 2.7. Davies studied the semiclassical complex harmonic oscillator =-=[5]-=-, and the general semiclassical Schrödinger equation with complex potential in one dimension [6]. He proved that spectral instability is generic, in the sense that the norm of the resolvent blows up a... |

2 | on the number of eigenvalues near the boundary of the pseudospectrum - Hager, Bound |

2 |
Linear Algebra and solvability of Partial Differential Equations
- Numerical
(Show Context)
Citation Context ...sense that the norm of the resolvent blows up as any power of semiclassical parameter h almost everywhere in the numerical range of the semiclassical principal symbol (see Theorem 2.1 below). Zworski =-=[74, 75]-=- made the important observation that the spectral instability of semiclassical differential operators is directly connected with the bracket condition and the solvability question. In these infinite d... |

1 | Operator Theory 53 - bounds, J - 2005 |

1 |
Structured pseudospectra and structured sensitivity of eigenvalues
- Du, Wei
(Show Context)
Citation Context ... use of the resolvent norm has given rise to the pseudospectral method in numerical analysis. There are several other ways of measuring spectral stability, for example the structured ε-pseudospectrum =-=[24]-=- and the second-order relative spectrum [41]. In the present review article, it will not be possible to make a more thorough treatment of the spectral instability of matrices. For more results, exampl... |

1 | spectrale semiclassique d’opérateurs non-autoadjoints - Instabilité |

1 |
On the completeness of the system of eigen- and associated functions of a non-selfadjoint differential operator
- Lidskiǐ
- 1956
(Show Context)
Citation Context ...of matrices. For more results, examples and references we refer the reader to [72]. Spectral instability also occurs for non-selfadjoint partial differential operators. Lidskiǐ, in a series of papers =-=[43]-=-–[53], studied the completeness and summability of eigenfunction expansions of certain types of non-selfadjoint differential operators. The first to study the stability of the spectrum for non-selfadj... |

1 | operators with a trace - Non-selfadjoint - 1959 |

1 |
pseudospectral behaviour for semiclassical operators in one dimension
- Boundary
(Show Context)
Citation Context ...ncipal type scalar semiclassical operators, there are still many open questions about the pseudospectrum. Pravda-Starov has studied the pseudospectrum for nonprincipal type operators in one dimension =-=[62]-=-, in the case when the Hessian of the principal symbol is elliptic and non-normal. He has also studied the pseudospectrum for general elliptic non-normal quadratic Weyl operators [63]. Sjöstrand and H... |

1 | for a class of semi-classical operators - Pseudo-spectrum |

1 |
Universality of the ESD and the Circular law (with an appendix by M
- Tao, Vu
(Show Context)
Citation Context ...d that for certain random perturbations of pseudodifferential operators, the spectrum will satisfy a asymptotic Weyl law, see [31]. It is interesting to compare this to the recent proof by Tao and Vu =-=[70]-=- of the circular law for random matrices, for which the spectrum is uniformly distributed in a disk. 3. Systems of semiclassical operators In this section, we will show how the results for semiclassic... |

1 |
Pseudospectra of semi-classical operators, talk at King’s
- Zworski
- 2001
(Show Context)
Citation Context ...h that (2.17) { z : |z − z0| ≤ c0(h log h −1 ) k k+1 } ∩ Spec(P(h)) = ∅ 0 < h ≤ h1 We obtain (2.17) from [22, Theorem 1.4] and (2.17) from [68]. In one dimension, the resolvent estimate was proved in =-=[73]-=-, and in some special cases by Boulton [3] who also showed that the bounds are optimal. As was demonstrated by Trefethen [71] this is also easy to see numerically. We have the following simple higher ... |