Wednesday, 10 September 2008

Eigenvalues of an element in a Weyl algebra

I have an operator acting on the polynomial algebra mathbbC[x,y,z] that I would like to find the eigenvalues/eigenvectors of. More specifically, let P(x1,ldots,x6) be a homogeneous polynomial, my operator has the form P(x,y,z,fracpartialpartialx,fracpartialpartialy,fracpartialpartialz). Are there any general strategies that could help me? For instance, say my operator were: z2yfracpartial3partialx2partialy+y3zfracpartial4partialy2partialz2+xz2fracpartial3partialxpartialz2.

My actual operator is degree 6 and more complicated, but other than that the same type of object. Any thoughts or references on how to attack this type of problem will be very welcome. Thanks a lot!



EDIT: My first example-operator was very poorly chosen, since every monomial would automatically be an eigenvector. I have now altered it a little to avoid this. Keep in mind this was only an example to show what type of object I am considering, and I am looking for general strategies. None the less, thanks for the quick response.



EDIT 2: I had misread the degree of my actual operator - it is 6.

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