How to compare DateTime in Extbase repository

Use string value of the date:

public function findNext() {
    $query = $this->createQuery();
    $date = new \DateTime('midnight');
    $query->matching(
        $query->greaterThanOrEqual('datum_beginn', $date->format('Y-m-d H:i:s'))
    );

    return $query->execute();
}