Spectrum and pseudospectrum of non-self-adjoint Schrödinger operators with periodic coefficients
The authors consider the pseudospectrum of the non-selfadjoint operator \[ D=-h^2\frac{d^2}{dx^2}+iV(x), \] where \(V(x)\) is an aperiodic entire analytic functions real on the real axis, with a real period \(T\). The operator \(h\) is treated as a small parameter. The authors 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 pseudo-eigenfunctions, i.e., the functions \(\varphi(h,x)\) satisfying the condition \[ \|D\varphi-\lambda\varphi\|=O(h^n),\quad \|\varphi\|= 1,\;n\in\mathbb N, \] can be constructed explicitly. The main result of the paper is as follows. Theorem. For a non-constant \(V(x)\) and for any positive integer \(n\), the \(h^n\)-pseudo-spectrum of the operator \(-h^2\frac{d^2} {dx^2}+iV(x)\), given on a circle, is the half-strip \([0,\infty)+i[\min V, \max V]\). Thus, it turns out that the pseudospectrum of the operator under study is much wider than its spectrum.
scientific article; zbMATH DE number 5155533