Physical modelling

2.1. The linear Schrödinger equation. We consider the situation of a single particle described by a selfadjoint Schrödinger operator $H_0=-\Delta+V: D\left(H_0\right) \subset L^2\left(\mathbb{R}^d\right) \rightarrow L^2\left(\mathbb{R}^d\right)$ with static pinning potential $V$. Apart from the static pinning potential, we also allow the presence of an additional control potential $V_{\text {con }}$ with time-dependent control function $u \in W_{\mathrm{pcw}}^{1,1}(0, T)$ (piecewise $W^{1,1}$ ). Thus, writing $V_{\mathrm{TD}}(t):=u(t) V_{\text {con }}$ for the time-dependent potential, we cover in this article time-dependent linear Schrödinger equations of the form

\[\begin{aligned} i \partial_t \psi(x, t) & =\left(H_0+V_{\mathrm{TD}}(x, t)\right) \psi(x, t), \quad(x, t) \in \mathbb{R}^d \times(0, T) \\ \psi(\bullet, 0) & =\varphi_0 \end{aligned}\]

The Schrödinger equation (2.1) with linear control potential appears naturally in the study of static physical systems, described by Schrödinger operators $H_0$, under the influence of a time-dependent electric field. This includes the study of the Stark effect that is the response of an atom or molecule to an external homogeneous constant electric field. In the so-called dipole approximation, the time-dependent control potential becomes $V_{\mathrm{TD}}=\langle p, E(t)\rangle$ where $p$ is the dipole moment of the object and $E$ the external (time-dependent) electric field acting on it.