Regularity of the inversion problem for the Sturm-Liouville difference equation III. A criterion for regularity of the inversion problem

We consider a difference equation −h −2∆(2)yn + qn(h)yn = fn(h), n ∈ Z = {0, ±1, ±2, . . . }, (1) where h ∈ (0, h0], h0 is a fixed positive number, ∆(2)yn = yn+1 − 2yn + yn−1, n ∈ Z; f = {fn(h)}n∈Z ∈ Lp(h), p ∈ [1, ∞), Lp(h) = {f : kfkLp(h) < ∞}, kfk p Lp(h) = X n∈Z |fn(h)| ph, and 0 ≤ qn(h) < ∞, Xn k=−∞ qk(h) > 0, X∞ k=n qk(h) > 0, n ∈ Z. We obtain necessary and sufficient conditions under which assertions I) - II) hold together: I) for a given p ∈ [1, ∞), for any f ∈ Lp(h), (1) has a unique solution y = {yn(h)}n∈Z ∈ Lp(h) (regardless of h), and y = (Gf)(h) def = {(Gf)n(h)}n∈Z, (Gf)n(h) = P m∈Z Gn,m(h)fm(h)h, n ∈ Z. II) kykLp(h) ≤ c(p)kfkLp(h) for any f ∈ Lp(h). Here c(p) is an absolute positive constant, {Gn,m(h)}n,m∈Z is the difference Green function corresponding to (