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Polymer quantization, singularity resolution and the 1/r^2 potential
We present a polymer quantization of the -lambda/r^2 potential on the
positive real line and compute numerically the bound state eigenenergies in
terms of the dimensionless coupling constant lambda. The singularity at the
origin is handled in two ways: first, by regularizing the potential and
adopting either symmetric or antisymmetric boundary conditions; second, by
keeping the potential unregularized but allowing the singularity to be balanced
by an antisymmetric boundary condition. The results are compared to the
semiclassical limit of the polymer theory and to the conventional Schrodinger
quantization on L_2(R_+). The various quantization schemes are in excellent
agreement for the highly excited states but differ for the low-lying states,
and the polymer spectrum is bounded below even when the Schrodinger spectrum is
not. We find as expected that for the antisymmetric boundary condition the
regularization of the potential is redundant: the polymer quantum theory is
well defined even with the unregularized potential and the regularization of
the potential does not significantly affect the spectrum.Comment: 21 pages, LaTeX including 7 figures. v2: analytic bounds improved;
references adde