Resonances and Scattering by a Periodic Structure

We consider a Schrödinger operator on the real line with a super-exponentially decaying and oscillating potential \(V(x)=e^{-x^2}\big(a-b e^{2 \mathrm{i} \alpha x}\big)\), where \(a,b\in \mathbb C\setminus\{0\}\) and \(\alpha>0\) are parameters. Let \(k^2\) be a spectral parameter. On the complex plane of \(k\), we find four infinite vertical sequences of resonances of this operator and four finite sequences of resonances located along certain rays in the complex plane. We obtain asymptotic representations for the resonances located far from the origin. The leading terms in the representations are found explicitly, while the error terms are estimated uniformly in \(a\) and \(b\). For certain values of the parameters, on the complex plane of \(k^2\), the vertical sequences might turn into sequences located near the real line, and thus, probably might be interesting for applications in physics.

DOI 10.1134/S1061920825600497