I would like to understand, possibly with an explicit example, how the free Schroedinger evolution does not leave $L^1$ invariant.
More precisely: given $f\in L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$, it is standard that $e^{it\Delta}f\in L^2(\mathbb{R}^d)\cap L^\infty(\mathbb{R}^d)\cap C^0(\mathbb{R}^d)$; can one find such a $f$ so that $\|e^{i t \Delta}f\|_1\to+\infty$ as $t\to t_0$ ?