I noticed that if I run:
Simplify[DiracDelta[x+y] - DiracDelta[-(x+y)]]
Mathematica returns zero, as it should since the DiracDelta is an even "function" (distribution). However, if I instead run:
Simplify[DiracDelta[x-y] - DiracDelta[-(x-y)]]
it does not simplify to zero. My first question is: is there a good reason why not (i.e., some weird scenario in which this wouldn't evaluate to zero under a well-defined integral)?
Secondly, is there an simple way to convince Mathematica to simplify expressions of this form, using the fact that the DiracDelta is even? (For example, I started creating a rule to tell Mathematica to use the evenness of the DiracDelta to make it always have the form delta[x1...] instead of delta[-x1...], and then simplify can do its work. But may be there are better ideas?)