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For a two-parameter family of Jacobi matrices exhibiting first-order spectralphase transitions, we prove discreteness of the spectrum in the positive realaxis when the parameters are in one of the transition boundaries. To this endwe develop a method for obtaining uniform asymptotics, with respect to thespectral parameter, of the generalized eigenvectors. Our technique can beapplied to a wide range of Jacobi matrices.

Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis