Adiabatic evolution generated by a Schrödinger operator with a continuous spectrum. II

V. A. Sergeev

Chebyshev Laboratory, St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We consider the non-stationary one-dimensional Schrödinger equation, the potential being a potential well linearly shrinking with time, with a small parameter $\varepsilon$ in front of the derivative with respect to time $\tau$. The solution to this equation that we study, $\Psi$, depends on the parameter $E$, which takes values in the continuous spectrum of the stationary Schrödinger operator. For a fixed $\tau$, the solution $\Psi$ is close to the generalized eigenfunction of the continuous spectrum of the stationary operator. We obtain for $\Psi$ asymptotics outside the potential well as $\varepsilon \to 0$.

Key words and phrases: non-stationary Schrödinger equation, continuous spectrum, time-dependent potential, adiabatic approximation, scattering of a plane wave by a sector with semi-transparent boundary.

UDC: 51.73

Received: 04.10.2025