Sharpness and necessity of regularity assumptions in power-asymptotics theorems

Determine whether the regularity assumptions used in the power-asymptotics results for Jacobi parameters—specifically the requirement of multi-term power expansions in the critical case—are close to optimal, or whether they can be significantly weakened while preserving the conclusions on limit circle vs limit point classification and the order/type of the Nevanlinna matrix.

Background

Theorems on Jacobi matrices with power asymptotics (including the critical case where |y_0| = 2x_0) provide near-complete classification and explicit orders/types under explicit multi-term expansions. It is unknown whether these regularity hypotheses can be relaxed.

Clarifying minimal regularity would broaden the applicability of these results and could illuminate borderline behaviours where diagonal and off-diagonal entries are nearly comparable.