Abstract

We study connections between continued fractions of type $J$ and spectral properties of second order difference operators with complex coefficients. It is known that the convergents of a bounded $J$--fractions are diagonal Padé approximants of the Weyl function of the corresponding difference operator, and that a bounded $J$--fraction converges uniformly to the Weyl function in some neighborhood of infinity. In this paper we establish convergence in capacity in the unbounded connected component of the resolvent set of the difference operator, and specify the rate of convergence. Furthermore, we show that the absence of poles of Padé approximants in some subdomain implies already locally uniform convergence. This enables us to verify the Padé conjecture for a subclass of Weyl functions.

For establishing these convergence results, we study the ratio and the $n$th root asymptotic behavior of Padé denominators of bounded $J$--fractions, and give relations with the Green function of the unbounded connected component of the resolvent set. In addition, we show that the number of ``spurious'' Padé poles in this set may be bounded.