  An eigenvalue problem relevant for non-linear sigma model with singular
metric is considered. We prove the existence of a non-degenerate pure point
spectrum for all finite values of the size R of the system. In the infrared
(IR) regime (large R) the eigenvalues admit a power series expansion around IR
critical point R\to\infty. We compute high order coefficients and prove that
the series converges for all finite values of R. In the ultraviolet (UV) limit
the spectrum condenses into a continuum spectrum with a set of residual bound
states. The spectrum agrees nicely with the central charge computed by the
Thermodynamic Bethe Ansatz method
