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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: $$ -\nabla(p(t)\Delta x(t))+q(t)x(t)=\lambda w(t)x(t),\quad t\in I, \tag1$$ where $I$ is the integer set $\{t\}_{t=a}^b$, $a$ is a finite integer or $-\infty$ and $b$ is a finite integer or $+\infty$ with $b-a\geq 3$; $\Delta$ and $\nabla$ are the forward and backward difference operators, respectively; $p(t)$ and $q(t)$ are all real-valued with $p(t)\neq 0$ for $t\in I$, $p(a-1)\neq 0$ if $a$ is finite, and $p(b+1)\neq 0$ if $b$ is finite; $w(t)>0$ for $t\in I$; and $\lambda$ 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.

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