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Self-adjoint extensions for second-order symmetric linear difference equations. (English) Zbl 1210.39004

Characterizations of self-adjoint extensions are fundamental in the study of spectral problems for symmetric linear differential and difference equations.

In this paper the authors consider self-adjoint extensions for the following second-order symmetric linear difference equation:

-(p(t)Δx(t))+q(t)x(t)=λw(t)x(t),tI,(1)

where I is the integer set {t} t=a b , a is a finite integer or - and b is a finite integer or + with b-a3; Δ and are the forward and backward difference operators, respectively; p(t) and q(t) are all real-valued with p(t)0 for tI, p(a-1)0 if a is finite, and p(b+1)0 if b is finite; w(t)>0 for tI; and λ is a complex spectral parameter.

The main tool used by the authors is based on the Glazman-Krein-Naimark theory for Hermitian subspaces. The authors study self-adjoint subspaces extensions and self-adjoint operator extensions of the minimal operators corresponding to equations (1), and they give a complete characterization of them in terms of boundary conditions, where the endpoints may be finite or infinite.

MSC:
39A12Discrete version of topics in analysis
39A70Difference operators
39A06Linear equations (difference equations)
47B15Hermitian and normal operators
47B25Symmetric and selfadjoint operators (unbounded)
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