A bound on the pseudospectrum for a class of non-normal Schrödinger operators.

Abstract

We are concerned with the non-normal Schrodinger 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 e-pseudospectrum of H will have an unbounded component for any e>0e>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 e>0e>0 such that the e-pseudospectrum σe(H)⊂{z:Rez≥R}∪⋃λ∈σ(H){z:∣∣z−λ∣∣<δ}.σe(H)⊂{z:Rez≥R}∪⋃λ∈σ(H){z:∣z−λ∣<δ}. In particular, the unbounded part of the pseudospectrum escapes towards +∞+∞ as e decreases. In addition, we give two examples of non-selfadjoint Schrodinger operators outside of our class and study their pseudospectra in more detail.