Mapping properties of the Schrödinger semigroup

Question

The Schrödinger semigroup $e^{t(-\Delta +V(x))}$ for Kato class potentials is fairly well-understood. A classical reference is the AMS paper "Schrödinger Semigroups" by Barry Simon. I was wondering what is known about the mapping property of $e^{i (-\Delta+V(x))}$ for $V(x)$ a confining potential, i.e. $V:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfies $\lim_{\vert x\vert \rightarrow \infty} V(x)=\infty$.

The operator $e^{i (-\Delta+V(x))}$ is (with a suitable choice of domain) unitary on $L^2(\mathbb{R}^n)$. I was wondering whether it is known for which $p,q$ we have that $e^{i (-\Delta+V(x))}$ is bounded from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$. I would assume that this type of question has been studied, but I have been unable to locate a suitable paper/book. Any pointers would be appreciated.

Answer 1

Partial answer: Let $p,q\in [1,\infty)$. In general one cannot expect this to be bounded for any combination of $p\neq q$. For a suitable choice of $c>0$ we have that the $L^2$ spectrum of $A=-\Delta+cx^2$ is contained in $2\pi \mathbb{N}_0$ and the $L^2$ eigenfunctions are given by Hermite functions. Furthermore, the Hermite functions are dense in $L^p(\mathbb{R})$ for $p\in [1,\infty)$ (see for example Theorem $6$ in Christian Berg, J. P. Reuschristensen, Density questions in the classical theory of moments, Annales de l’institut Fourier, tome 31, no 3 (1981), p. 99-114 or Theorem $10.7$ in "Gaussian Harmonic Analysis" by Wilfredo Urbina-Romero). Thus, for suitable $c$ we get that $e^{i(-\Delta+cx^2)}$ is the identity on $L^p(\mathbb{R})$, for $p\in [1,\infty)$, and hence will not be bounded for any $q\neq p$. The same example works for all other $n$ too.

It's still to be understood what happens for $p=\infty$.