Spectrum and pseudospectrum of non-self-adjoint Schrödinger operators with periodic coefficients

We consider the pseudospectrum of the non-self-adjoint operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{D} = - h^2 \frac{{d^2 }}{{dx^2 }} + iV(x)$$ \end{document}, where V(x) is a periodic entire analytic function, real on the real axis, with a real period T. In this operator, h is treated as a small parameter. We show that the pseudospectrum of this operator is the closure of its numerical image, i.e., a half-strip in ℂ. In this case, the pseudoeigenfunctions, i.e., the functions ϕ(h, x) satisfying the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\| {\mathfrak{D}\varphi - \lambda \varphi } \right\| = O(h^N ), \left\| \varphi \right\| = 1, N \in \mathbb{N}$$ \end{document}, can be constructed explicitly. Thus, it turns out that the pseudospectrum of the operator under study is much wider than its spectrum.