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:
where is the integer set , is a finite integer or and is a finite integer or with ; and are the forward and backward difference operators, respectively; and are all real-valued with for , if is finite, and if is finite; for ; and 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.