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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:

$-\nabla \left(p\left(t\right){\Delta }x\left(t\right)\right)+q\left(t\right)x\left(t\right)=\lambda w\left(t\right)x\left(t\right),\phantom{\rule{1.em}{0ex}}t\in I,\phantom{\rule{2.em}{0ex}}\left(1\right)$

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

MSC:
 39A12 Discrete version of topics in analysis 39A70 Difference operators 39A06 Linear equations (difference equations) 47B15 Hermitian and normal operators 47B25 Symmetric and selfadjoint operators (unbounded)
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