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We study spectrum of finite truncations of unbounded Jacobi matrices withperiodically modulated entries. In particular, we show that under somehypotheses a sequence of properly normalized eigenvalue counting measuresconverge vaguely to an explicit infinite Radon measure. To do so we link theasymptotic behavior of the Christoffel-Darboux kernel on the diagonal with thelimiting measure. Finally, we derive strong asymptotics of the associatedorthogonal polynomials in the complex plane, which allows us to prove thatCauchy transforms of the normalized eigenvalue counting measures convergepointwise and which leads to a stronger notion of convergence.

Asymptotic zeros' distribution of orthogonal polynomials with unbounded recurrence coefficients