Rapidly Decaying Wigner Functions are Schwartz Functions

Abstract

We show that if the Wigner function of a (possibly mixed) quantum state decays toward infinity faster than any polynomial in the phase space variables $x$ and $p$, then so do all of its derivatives, i.e., it is a Schwartz function on phase space. This is equivalent to the condition that the Husimi function is a Schwartz function, that the quantum state is a Schwartz operator in the sense of Keyl et al., and, in the case of a pure state, that the wavefunction is a Schwartz function on configuration space. We discuss the interpretation of this constraint on Wigner functions and provide explicit bounds on Schwartz seminorms.

References in corpus (2)

• Schwartz operators
• Quantum Bochner's theorem for phase spaces built on projective representations

Cited by in corpus (1)

• Classical correspondence beyond the Ehrenfest time for open quantum systems with general Lindbladians