Unusual poles of the $ζ$-functions for some regular singular differential operators
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We consider the resolvent of a system of first order differential operators with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents powers of $λ$ which depend on the singularity, and can take even irrational values. The consequences for the pole structure of the corresponding $ζ$ and $η$-functions are also discussed.
26 pages, 1 figure, LaTeX. References added. Version to appear in the Journal of Physics A: Math. and Gen
26 pages, 1 figure, LaTeX. References added. Version to appear in the Journal of Physics A: Math. and Gen