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"Eigenvalue spectroscopy." How do you like this method?

Assume that you have two computer programs, one being for solving the initial value problem for one-dimensional quantum mechanical system, the other being for doing Fourier analysis.  Assume an appropriate static potential function like a parabolic function V(x)=x^2.  And set an appropriate initial wave function PSI(x,0) for the initial value problem program.

Now, let's go!  First you let the initial value problem program run for a certain period, during which you also record the real part value PSIR(xs,t) and imaginary part value PSII(xs,t) of the wave function at a particular fixed sampling point xs.  So you can obtain a couple of sequential data PSIR(t) and PSII(t).  Next you perform the Fourier analysis on these data, and you draw a Fourier transform power function FR(w)^2+FI(w)^2 (w for Greek lower case omega, the angular frequency.)

Can you predict the result of the Fourier analysis?

Hey, presto!  It's the eigenvalue spectrogram!

Of course, the result depend on what function you set as the initial wave function PSI(x,0).  If you choose an even fucntion like cos(3x)exp(-x^2/s^2) as the initial function, then you will get only even order peaks in the spectrogram (see the attached file EVENCASE.docx.)  Conversely if you choose an odd function like sin(3x)exp(-x^2/s^2), then you will get only odd order peaks(file ODDCASE.docx.)

Thus far, I have tried only a few examples.  I cannot still prove the mechanism of this method mathematically.  Can anybody prove this?

Thank you for reading

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