What is the Interpretation of the Inverse Property of a Delta Function in QM?

[Peeter]

In a book on QM are listed a few properties of the delta function, one of which is:

$$x \delta^{-1}(x) = - \delta(x)$$

I can't figure out how to interpret that? Putting the statement in integral form isn't particularily enlightening looking:

$$f(x) = \int f(x-x') \delta(x') dx' = \int -x' \delta^{-1}(x') f(x - x') dx'$$

any hint what this property is about or how one would show it?

 

[Preno]

I'm not aware of any meaningful operation corresponding to the "inverse of the delta function" in regular distribution theory and even intuitively speaking, it doesn't seem to be meaningful. Are you sure it's not actually:

$x\delta^\prime = -\delta,$

which is a meaningful and true identity?

 

[Peeter]

Thanks Preno. Your statement makes sense (ie: can show it with integration by parts).

I'm pretty sure it was listed as ^{-1}, but will have to wait til I'm home to verify. It's probably another typo in the text.

Confirmed. Just another typo in the text.