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Suppose I have a Wigner function that falls off faster than any polynomial for all directions in phase space. That is, for all $a,b>0$, $$\lim_{|x|\to\infty} |x^a p^b W(x,p)| =0=\lim_{|p|\to\infty} |x^a p^b W(x,p)|,$$ which, when combined with the boundedness of its magnitude, is known to be equivalent to $|W|_{(a,b),(0,0)} < \infty$ where $$|W|_{(a,b),(c,d)}\equiv \sup_{x,p} |x^a p^b \partial_x^c \partial_p^d W(x,p)|$$ using some standard tricks in Fourier analysis. Does this means $W$ is a Schwartz function, i.e., that $|W|_{(a,b),(c,d)} < \infty$ for all $a,b,c,d$?

This of course is not true for a generic function on phase space, but I am pretty sure it holds for Wigner functions corresponding to the density matrix (a positive operator). Intuitively, the Wiger function's high-frequency wiggles in the $x$ direction correspond to coherence over long distances in $p$, but if the support of (most of) the Wigner function's mass is basically bounded in $p$, these $x$ wiggles must be suppressed above some frequency cut-off. And vice versa.

I thought I remember this being demonstrated in Folland's Harmonic Analysis in Phase Space, but I can't dig it up even after a lot of searching.

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    $\begingroup$ Not sure if you'd find this useful... Are the books of Schleich or L Cohen interested? $\endgroup$ Sep 1, 2020 at 15:41
  • $\begingroup$ Thanks! Based on the intro, that paper seems like it must have the answer buried inside somewhere. Any chance you think you know which section I should concentrate on? They develop a lot of formalism over 50 pages, so it will take me a while to work through it all. $\endgroup$ Sep 1, 2020 at 15:52
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    $\begingroup$ Sorry, I could not bring myself to put a finger on it, any more than I could take Folland Ch 3... Searching for "Wigner" in that paper might get you warm.... $\endgroup$ Sep 1, 2020 at 15:56
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    $\begingroup$ Here, further, is Wong's concise math book on the stuff but it's been left at the COVID isolated office... Frankly, e.g for the oscillator, all Wiggies are built out of the ground state Gaussian acted on by polynomials, so the statement is trivially valid, but I understand you seek rigorous backup... $\endgroup$ Sep 3, 2020 at 14:24

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Felipe Hernández and I ended up showing that the answer is yes for both pure and mixed states: "Rapidly Decaying Wigner Functions are Schwartz Functions" [arXiv:2103.14183].

We were able to laboriously find an explicit bound (Theorem 3.9 in v1 of the arXiv paper): $$|W|_{(a,b),(c,d)} \le (2\pi)^{5n} 2^{4(|a|+|b|+|c|+|d|+n)} \|W_\chi\|_{(a,b),(c,d)} \|W_\chi\|_{(2a+2d+12,2b+2c+12),(0,0)}$$ $$\qquad\times \|W\|_{(2a+2d+8,2b+2c+8),(0,0)}.$$ Here, $W_\chi$ is the Wigner function for an arbitrary Schwartz function (wavepacket) $\chi$ (which can be taken to be, e.g., Gaussian), $n$ is the dimension, $a=(a_1,\ldots,a_n)$ is a multi-index, $|a|=\sum_{i=1}^n a_i$ is the multi-index norm, $|W|_{(a,b),(c,d)} = \sup_{x,p} |x^a p^b \partial_x^c \partial_p^d W(x,p)|$ is the Schwartz semi-norm (with, e.g., $x^a = x_1^{a_1}\cdots x_n^{a_n}$ and $\partial_p^d =\partial_{p_1}^{d_1}\cdots \partial_{p_n}^{d_n}$), and $$\|W\|_{(a,b),(c,d)} = \sum_{a'\le a}\sum_{b'\le b}\sum_{c'\le c}\sum_{d'\le d} |W|_{(a',b'),(c',d')}$$ the Schwartz norm (with $a\le a'$ meaning $a_i \le a_i^\prime$ for all $i=1,\ldots,n$).


(EDIT: After posting the preprint, we were alerted to an earlier theorem by K. Gröchenig that superficially seemed to prove this in the pure case, but a more careful reading revealed that Gröchenig's theorem relied on important additional assumptions, unlike our results. See Theorem 11.2.5 and its Corollary 11.2.6, on pages 228-229 of K. Gröchenig, "Foundations of Time-frequency Analysis" (2001).)

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