Semiclassical spectral instability for non-self-adjoint operators. I: A model. (Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I: un modèle.) (French) Zbl 1131.34057

Hager, Mildred

Ann. Fac. Sci. Toulouse, Math. (6) 15, No. 2, 243-280 (2006).

A simple 1-dimensional model problem is discussed and explored. Let \(\partial\) denote 1-dimensional differentiation, then \(-\mathrm{i}\,\partial\) is considered as a selfadjoint operator on the interval \(\left[0,2\pi\right]\) with periodicity boundary conditions. Perturbations associated with this operator of the form \(P=-\mathrm{i}\, h\,\partial+g\left(m\right)\) are considered, where \(g\left(m\right)\) indicates the multiplication operator \(\left(g\left(m\right)f\right)\left(x\right)=g\left(x\right)\, f\left(x\right)\), \(x\in\left[0,2\pi\right]\), for an analytic complex-valued function \(g\) with \(g^{\prime}\) having a nonvanishing imaginary part and \(h\in]0,1]\) is a parameter. Whereas the spectrum of \(-\mathrm{i}\, h\,\partial\) is simply \(\frac{2\pi}{h}\left[\mathbb{Z}\right]\), it is shown that for the perturbed case the eigenvalues of \(P\) are distributed within the semi-classical “pseudo”-spectrum (given as the closure of the range of the semi-classical symbol \(\left(x,\xi\right)\mapsto h\xi+g\left(x\right)\)) according to a two-dimensional Weyl law.

Reviewer: Rainer Picard (Dresden)

MSC:

34L05	General spectral theory of ordinary differential operators
47E05	General theory of ordinary differential operators
34D10	Perturbations of ordinary differential equations

Keywords:

non-selfadjoint operators; semi-classical symbol; semi-classical pseudospectrum

References:

[1]	Agmon, S., Lectures on elliptic boundary value problems (1965) · Zbl 0151.20203
[2]	Davies, E. B., Semiclassical states for Non-Self-Adjoint Schrödinger Operators, Commun. Math. Phys., 200, 35-41 (1999) · Zbl 0921.47060
[3]	Burq, N.; Zworski, M., Semi-Classical Propagation, 223, 1-12 (2001) · Zbl 1042.81582
[4]	Davies, E. B., Pseudospectra of differential operators, J.Operator theory, 43, 243-262 (2000) · Zbl 0998.34067
[5]	Davies, E. B., Semigroup growth bounds · Zbl 1114.47040
[6]	Hörmander, L., The analysis of Linear Partial Differential Operators, 1-3, 256-257 (19831985)
[7]	Levin, B. Ja., Distribution of Zeros of entire functions (1964) · Zbl 0152.06703
[8]	Reddy, S.; Schmid, P.; Henningson, D., Pseudospectra of the Orr-Sommerfeld operator, Siam J. Appl. Math., 53, 15-45 (1993) · Zbl 0778.34060
[9]	Sjöstrand, J.; Grigis, A., Microlocal Analysis for Differential Operators, 196 (1994) · Zbl 0804.35001
[10]	Sjöstrand, J.; Dimassi, M., Spectral Asymptotics in the Semi-Classical Limit, 268 (1999) · Zbl 0926.35002
[11]	Sjöstrand, J., Astérisque, 95 (1982) · Zbl 1384.35004
[12]	Sjöstrand, J.; Zworski, M.; Dencker, N., Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math., 57, 384-415 (2004) · Zbl 1054.35035
[13]	Sjöstrand, J., Lectures on resonances · Zbl 0877.35090
[14]	Tang, S. H.; Zworski, M., Resonance expansion of scattered waves, Comm. Pure Appl. Math., 53, 1305-1334 (2000) · Zbl 1032.35148
[15]	Titchmarsh, E. C., The theory of functions (1939) · JFM 65.0302.01
[16]	Trefethen, L. N., Numerical analysis, 260, 234-266 (1991)
[17]	Trefethen, L. N., Pseudospectra of linear operators, SIAM, 39, 3, 383-406 (1997) · Zbl 0896.15006
[18]	Trefethen, L. N., Wave packet Pseudomodes of variable coefficient differential operators, Proceedings of the Royal Society, Series A, 461, 3099-3122 (2005) · Zbl 1206.34109
[19]	Zworski, M., A remark on a paper of E.B. Davies, Proceedings of the AMS, 129, 2955-2957 (1999) · Zbl 0981.35107
[20]	Zworski, M., Numerical linear algebra and solvability of partial differential equations, Comm. Math. Phys., 229, 293-307 (2002) · Zbl 1021.35077