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

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