Geodesic
Spectrum and Pseudospectrum of non-self-adjoint Schr\"odinger Operators with Periodic Coefficients
Matematičeskie zametki, Tome 80 (2006) no. 3, pp. 356-366.

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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.
Keywords: spectrum, pseudospectrum, Schrödinger operator, periodicity condition, periodic entire analytic function, non-self-adjoint operator, Riemann surface. 
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     author = {S. V. Galtsev and A. I. Shafarevich},
     title = {Spectrum and {Pseudospectrum} of non-self-adjoint {Schr\"odinger} {Operators} with {Periodic} {Coefficients}},
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S. V. Galtsev; A. I. Shafarevich. Spectrum and Pseudospectrum of non-self-adjoint Schr\"odinger Operators with Periodic Coefficients. Matematičeskie zametki, Tome 80 (2006) no. 3, pp. 356-366. http://geodesic.mathdoc.fr/item/MZM_2006_80_3_a4/

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