Rapidly Decaying Wigner Functions are Schwartz Functions

TL;DR

This paper proves that quantum states with Wigner functions decaying faster than any polynomial are Schwartz functions, implying smoothness and rapid decay of all derivatives, with implications for quantum phase space analysis.

Contribution

It establishes the equivalence between rapid decay of Wigner functions and the Schwartz class property for quantum states, extending understanding of phase space representations.

Findings

Wigner functions decaying faster than polynomials are Schwartz functions

All derivatives of such Wigner functions also decay rapidly

Provides bounds on Schwartz seminorms for these 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.