03 Feb '06 00:112 edits

It can be shown that the equation x'' + (E - t^2)x = 0 has a solution which converges to 0 for large t if and only if E is an odd integer (I've just had to prove this bit on a coursework -- we can take that as read. In fact, the solution is, for E = 2k+1, P*exp(-t^2/2) where P is a certain kth-degree polynomial which is not that important here).

This suggests a question for the physicists, because that equation seems to describe an oscillator of some kind, although weirder than any I've dealt with. Can someone tell me how an oscillator like this would behave, or what its physical realisation might be?

Also, the bit that intrigued me was the fact that there might be a real, physical situation which can only be realised for certain

Typing the last paragraph gave me a nauseous feeling that this might somehow involve quantum mechanics, but I don't have the physics background to know for sure. Anyone?

This suggests a question for the physicists, because that equation seems to describe an oscillator of some kind, although weirder than any I've dealt with. Can someone tell me how an oscillator like this would behave, or what its physical realisation might be?

Also, the bit that intrigued me was the fact that there might be a real, physical situation which can only be realised for certain

*discrete*values of a parameter. The fact that E must be odd for the solution to converge comes from a recurrence relation between the coefficients in a series expansion of the solution, and it should be possible to generate equations like the one above where, rather than being odd, E must be of the form 9k+7 or 4k+2 (or something) in order for such a solution to exist. I wonder if these equations can also be interpreted physically.Typing the last paragraph gave me a nauseous feeling that this might somehow involve quantum mechanics, but I don't have the physics background to know for sure. Anyone?