The problem: regularity of the IDS.

In general, even for the simple model, very little known. 
E.g. it is not known that the IDS is a continuous function!

The physical intuition is that the IDS should benefit from regularity of 
the IDS of and of the "IDS" of the potential

Indeed, as  if background operator and potential would commute,



Actually, as the operators do not commute, one can even expect
more regularity for the IDS of the total operator.

Except for one dimensional operators, all the results on the regularity of 
the IDS use the ideas we now describe.