Pointwise bounds on eigenfunctions and wave packets in $N$-body quantum systems. I

Abstract

We provide a simple proof of (a modification of) Kato’s theorem on the Hölder continuity of wave packets in N-body quantum systems. Using this method of proof and recent results of O’Connor, we prove a pointwise bound \[ |\Psi (\zeta )| \leqq {D_\varepsilon }\exp [ - (1 - \varepsilon ){a_0}|x|]\] on discrete eigenfunctions of energy E. Here $\varepsilon > 0,a_0^2 = 2$ (mass of the system) $[{\text {dist}}(E,{\sigma _{{\text {ess}}}})]$ and $|x|$ is the radius of gyration.