#### SemM

Gold Member

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Regarding the two main operators, X and D, each has a domain in ##\mathscr{H}## which represent the set of admissible functions, and that satisfy the completeness relation in ##\mathscr{H}## and the inner product.

In practice in quantum chemistry or in quantum physics, one would like to know the form of these functions. In Bohms "Quantum Theory" Bohm shows an explicitely example of such a function ##f(x)=e^{-x^2}g(x)## where g is a polynomial. This is an ideal example of such a function, however in this book, as well as in others, it seems that these functions "are pulled out of thin air", without any method to it. Is this a trial and error process fueled by the experience of the mathematician to find such a function? There are, probably other forms than the above-given, so my question is, is there a method to find a list of functions in the the domain of an operator?

Thanks!