Spectrum and Pseudospectrum of non-self-adjoint Schr\"odinger Operators with Periodic Coefficients

S. V. Galtsev ; A. I. Shafarevich

Matematičeskie zametki, Tome 80 (2006) no. 3, pp. 356-366

Résumé

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.

Détail

Keywords: spectrum, pseudospectrum, Schrödinger operator, periodicity condition, periodic entire analytic function, non-self-adjoint operator, Riemann surface.