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An eigenvalue problem relevant for non-linear sigma model with singularmetric is considered. We prove the existence of a non-degenerate pure pointspectrum 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 IRcritical point R\to\infty. We compute high order coefficients and prove thatthe series converges for all finite values of R. In the ultraviolet (UV) limitthe spectrum condenses into a continuum spectrum with a set of residual boundstates. The spectrum agrees nicely with the central charge computed by theThermodynamic Bethe Ansatz methodComment: 18 pages, AmsLatex, Axodra

An eigenvalue problem related to the non-linear -model: analytical and numerical results