On Higher-Order Difference Operators

Abstract

For fixed p, q ≥ 1 we consider the equation

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadY % eacaWG5bGaaiykamaaBaaaleaacaWG2baabeaakiabggMi6oaaqaha % baGaamyyamaaBaaaleaacaWG2bGaamOAaiaadMhacaWG2bGaey4kaS % IaamOAaaqabaaabaGaamOAaiabg2da9iabgkHiTiaadghaaeaacaWG % WbaaniabggHiLdGccqGH9aqpcqaH7oaBcaWG5bWaaSbaaSqaaiaadA % haaeqaaOGaaiilaiaadAhacqGHLjYScaWGXbGaaiilaiaadggadaWg % aaWcbaGaamODaiaacYcacqGHsislcaWGXbaabeaakiabg2da9iaaig % dacaGGSaaaaa!5B05!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${(Ly)_v} \equiv \sum\limits_{j = - q}^p {{a_{vjyv + j}}} = \lambda {y_v},v \geqslant q,{a_{v, - q}} = 1,$$

(1)

where y = [y _v ] _v≥0 and a _vj ∈ ℂ are complex numbers. The operator L is called non-degenerate if a _vp ≠ 0, v ≥q. In this article we define the Weyl matrix for L, establish its properties, and consider the inverse problem of determining L from its Weyl matrix. We obtain an algorithm of solving, and necessary and sufficient conditions of solvability of the inverse problem, prove the uniqueness theorem. Analogous results are valid for (1) in an abstract space. We note that the inverse problem for (1) has applications in the theory of nonlinear difference equations (see e.g. [1, 2]). The case p = q = 1 in (1) was investigated in [3]–[5] and other works.