The role of the weight function for adjoint DO

[Jianphys17]

Hi at all, I've a curiosity about the role that the weight function w(t) she has, into the define of adjoint & s-adjoint op.
It is relevant in physical applications or not ?

 

[S.G. Janssens]

Can you give an example of what you are talking about? Am I correct that you are talking about weight functions appearing in the definition of an inner product on a space of (equivalence classes of) square integrable functions?

 

[Jianphys17]

Yes..

 

[S.G. Janssens]

Well, one could say that the weight factor determined for which functions the (weighted) $L^2$-norm is finite, hence which functions are admissible states for the system under consideration. It also determines which functions are orthogonal to each other. To me it seems that these properties (being an admissible state and being orthogonal to other admissible states) are quite relevant from a physical point of view.

To become more concrete, one would have to specify which physical system (or: class of systems) is under consideration.