Timeline for Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?
Current License: CC BY-SA 3.0
7 events
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| Aug 20, 2013 at 0:49 | comment | added | Selene Routley | @BielsNohr I think your comment to Christoph describes a very wise approach. I find many things in physics likewise: deductions are made with no proof that would withstand a mathematician's scrutiny. Often a "physically reasonable postulate" is implicitly being made: here we make an implicit assumption that it is physically reasonable to restrict our theory to observables whose eigenfunctions have inner products well behaved in the way you describe. I wish more authors were a bit more forthright about their assumptions - there's nothing wrong with making them: it is physics, after all. | |
| Oct 2, 2012 at 18:06 | comment | added | Qmechanic♦ | The title question(v1) concerns just the eigenvalue spectrum (= point spectrum), not the full spectrum. | |
| Aug 13, 2012 at 17:48 | comment | added | jjcale | @Qmechanics: Question (v1) is also non-trivial on standard Hilbert spaces, since not every value of the spectrum is an eigenvalue. So one has to prove that isolated points of the spectrum are indeed eigenvalues. | |
| Aug 12, 2012 at 19:58 | comment | added | Qmechanic♦ | Comment to the answer(v1): Most mathematical books on functional analysis would just treat (bounded and unbounded) operators as living on a (standard) Hilbert space (where the inner product exists by definition). OP's title question(v1) only becomes non-trivial if one goes beyond the framework of standard Hilbert spaces, e.g. in the context of rigged Hilbert spaces. | |
| Aug 12, 2012 at 9:31 | comment | added | BielsNohr | @Christoph also captured this in his post, and I would like to acknowledge that. All of these answers have helped me realize that the question I was asking does not have a simple answer like the author of the textbook I was reading implied; additionally, I think a full comprehension will require a few more years of study in math on my part. Thanks all. | |
| Aug 12, 2012 at 9:25 | vote | accept | BielsNohr | ||
| Aug 10, 2012 at 18:57 | history | answered | jjcale | CC BY-SA 3.0 |