Normalizable wave functions? 
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*How can I test whether a wave function is normalizable?

*If you apply an operator to a wave function, sometimes the result will not be normalizable. But how can I find these wave functions that do not correspond to normalizable eigenstates of this operator?

*The Hamilton operator for the harmonic oscillator, I am told, has both a discrete and a continuous spectrum. The discrete spectrum is the eigenvalues, but how do I find the continuous spectrum? It's relevant because I am also told these are the non normalizable values of the spectrum.

 A: *

*You test a wave function for normalizability by integrating its square magnitude. If you get a finite result then it is normalizable. To spare you complicated integrations you can also take a simpler wave function that you know is normalizable and compare it using the usual arguments.

*An operator is not only defined by the mathematical operation it performs, but also by which space it acts on. The Hilbert space of square integrable functions is where quantum operators act on. So by definition they take a square integrable function and give you a square integrable function. It can happen though that such an operator has eigenfunction that are not in the set of square integrable functions. For some of these cases consistency requires that we extend the hilbert space (or its dual) we work with. The eigenfunctions of the position and momentum operators fall into this category. See the Gelfand construction or rigged Hilbert space for details.

*The hamiltonian of the harmonic oscillator as an operator on the quantum hilbert space has only a discrete spectrum. If you define it to be an operator on a more general space that admits functions that are not square integrable then the spectrum may in fact be continuous. This is the perfect example for why an operator must always be stated with the space it acts on.
