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

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

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