One of the Sturm-Liouville oscillation theorems (say, in its
Kneser-type formulation for Schroedinger differential operator in
one dimension) states that the inverse quadratic asymptotic
potentials are critical and separate the asymptotically
non-oscillatory and oscillatory behavior of wave-functions. Paper
describes an extension of this theorem to its difference-equation
Jacobi analogue. The generalization is interesting, being
perceivably less easy due to the breakdown of symmetry of the
discrete version of the relevant operator. Still, a satisfactorily
close analogy of the new theorem to its differential-limit
predecessor is preserved.



MR2049680 Luef, Franz ; Teschl, Gerald . On the finiteness of the
number of eigenvalues of Jacobi operators below the essential
spectrum. J. Difference Equ. Appl. 10 (2004), no. 3, 299--307.