Spectrum and Pseudospectrum of non-self-adjoint Schrödinger Operators with Periodic Coefficients

We consider the pseudospectrum of the non-self-adjoint operator
$$ \mathfrak D=-h^2\frac{d^2}{dx^2}+iV(x), $$
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 $\mathbb C$. In this case, the pseudoeigenfunctions, i.e., the functions $\varphi(h,x)$ satisfying the condition
$$ \|\mathfrak D\varphi-\lambda\varphi\|=O(h^N), \qquad \|\varphi\|=1, \quad N\in\mathbb N, $$
can be constructed explicitly. Thus, it turns out that the pseudospectrum of the operator under study is much wider than its spectrum.