A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

Dondl, P.; Dorey, P.; Rossler, F.

doi:10.1093/amrx/abw011

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

Dondl, P.; Dorey, P.; Rossler, F.

Authors

P. Dondl

Professor Patrick Dorey p.e.dorey@durham.ac.uk
Professor

F. Rossler

Abstract

We are concerned with the non-normal Schrödinger operator H=−Δ+VH=−Δ+V on L2(Rn)L2(Rn) , where V∈W1,∞loc(Rn)V∈Wloc1,∞(Rn) and ReV(x)≥c∣x∣2−dReV(x)≥c∣x∣2−d for some c,d>0c,d>0 . The spectrum of this operator is discrete and its real part is bounded below by −d−d . In general, the ε-pseudospectrum of H will have an unbounded component for any ε>0ε>0 and thus will not approximate the spectrum in a global sense. By exploiting the fact that the semigroup e−tHe−tH is immediately compact, we show a complementary result, namely that for every δ>0δ>0 , R>0R>0 there exists an ε>0ε>0 such that the ε-pseudospectrum σε(H)⊂{z:Rez≥R}∪⋃λ∈σ(H){z:∣∣z−λ∣∣<δ}.σε(H)⊂{z:Rez≥R}∪⋃λ∈σ(H){z:∣z−λ∣<δ}. In particular, the unbounded part of the pseudospectrum escapes towards +∞+∞ as ε decreases. In addition, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail.

Citation

Dondl, P., Dorey, P., & Rossler, F. (2017). A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators. Applied mathematics research express, 2017(2), 271-296. https://doi.org/10.1093/amrx/abw011

Journal Article Type	Article
Acceptance Date	Nov 10, 2016
Online Publication Date	Dec 21, 2016
Publication Date	Sep 1, 2017
Deposit Date	Mar 29, 2017
Publicly Available Date	Dec 21, 2017
Journal	Applied Mathematics Research eXpress
Print ISSN	1687-1200
Electronic ISSN	1687-1197
Publisher	Hindawi
Peer Reviewed	Peer Reviewed
Volume	2017
Issue	2
Pages	271-296
DOI	https://doi.org/10.1093/amrx/abw011
Related Public URLs	https://arxiv.org/pdf/1505.05719.pdf

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Copyright Statement
This is a pre-copyedited, author-produced version of an article accepted for publication in Applied Mathematics Research eXpress following peer review. The version of record Dondl, P., Dorey, P. & Rossler, F. (2017). A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators. Applied Mathematics Research eXpress 2017(2): 271-296 is available online at: https://doi.org/10.1093/amrx/abw011.