# A remark on the attainable set of the Schr\"odinger equation

## TL;DR

This paper investigates the set of wavefunctions reachable through the Schrödinger equation with variable potentials, demonstrating that this set has no interior points in the relevant function space, indicating limitations in controllability.

## Contribution

It provides a mathematical analysis showing the attainable set of wavefunctions under Schrödinger dynamics with variable potentials has empty interior, revealing fundamental constraints.

## Key findings

- The attainable set has empty interior in the $L^2$ sphere.
- The set of trajectories also has empty interior.
- Implications for controllability of quantum systems.

## Abstract

We discuss the set of wavefunctions $\psi_V(t)$ that can be obtained from a given initial condition $\psi_0$ by applying the flow of the Schr\"odinger operator $-\Delta + V(t,x)$ and varying the potential $V(t,x)$. We show that this set has empty interior, both as a subset of the sphere in $L^2(\mathbb{R}^d)$ and as a set of trajectories.