Discrete spectrum in a critical coupling case of Jacobi matrices with   spectral phase transitions by uniform asymptotic analysis
  For a two-parameter family of Jacobi matrices exhibiting first-order spectral phase transitions, we prove discreteness of the spectrum in the positive real axis when the parameters are in one of the transition boundaries. To this end we develop a method for obtaining uniform asymptotics, with respect to the spectral parameter, of the generalized eigenvectors. Our technique can be applied to a wide range of Jacobi matrices. 
Serguei Naboko, Irina Pchelintseva, Luis O. Silva