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Classification of non-self-adjoint singular Sturm-Liouville difference equations. (English) Zbl 1228.39006
This paper is concerned with the Sturm-Liouville difference equation with complex coefficients: $$-\nabla \left( p(n)\Delta x(n)\right) +q(n)x(n)=\lambda w(n)x(n),\ n\in {\mathbb{N}}_{0}=\{0,1,2,\dots\}, \tag1$$ where $\left\{ p(n)\right\} _{n=-1}^\infty, \left\{q(n)\right\}_{n=0}^{\infty}$ are complex sequences such that $p(n)\ne 0$ for $n=0,1,2,\dots$, $\left\{ w(n)\right\} _{n=0}^{\infty }$ is a positive real sequence, and $\lambda$ is a spectral parameter. An important question is whether a solution $\left\{ x(n)\right\} _{n=-1}^{\infty }$ (which depends on $\lambda$) is square summable, that is, whether the series $\sum_{n=0}^{\infty }w(n)\left\vert x(n)\right\vert ^{2}$ is finite. Let $$\Omega =\overline{co}\left\{ \frac{q(n)}{w(n)}+rp(n),\ n\in {\mathbb{N}}_{0},\ 0<r<\infty \right\},$$ where $\overline{co}$ stands for the closed convex hull, let $$S=\left\{ (\theta ,K):K\in \partial \Omega ,\text{Re}\left\vert (z-K)e^{i\theta }\right\vert \ge 0\text{ for all }z\in \Omega \right\},$$ and let $$\Lambda _{\theta ,K}=\left\{ \lambda \in {\mathbb{C}}:\text{Re}\left( \left( \lambda -K\right) e^{i\theta }\right) <0\right\}.$$ The authors give the following definition: Let $(\theta ,K)\in S$ and $\lambda \in \Lambda _{\theta,K}$. If (1) has exactly one linearly independent solution $\left\{x(n)\right\}$ satisfying $$\sum_{n=0}^{\infty }\text{Re}\left[ e^{i\theta }\left( q(n)-Kw(n)\right) \right] \left\vert x(n)\right\vert ^{2}+\sum_{n=0}^{\infty }\text{Re}\left( e^{i\theta }p(n)\right) \left\vert \Delta (n)\right\vert ^{2}+\sum_{n=0}^{\infty }w(n)\left\vert x(n)\right\vert ^{2}<\infty, \tag2$$ and this is the only linearly independent square summable solution, then (1) is of Type I. If (1) has exactly one linearly independent solution satisfying (2), but all solutions of (1) are square summable, then (1) is of Type II. If all solutions of (1) satisfy (2) (and hence square summable), then (1) is of Type III. It is shown that a Type I equation does not depend on $\Lambda _{\theta ,K},$ but Type II and Type III equations do. The results in this paper supplement those in an earlier paper by {\it R. H. Wilson} [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2057, 1505--1531 (2005; Zbl 1145.47303)]. In particular, the dependence of Type II and Type III equations on $\Lambda _{\theta ,K}$ is discussed. Remark: Since $p(-1)$ is also involved in (1), it may seem necessary to explain why this number does not play any role in $\Omega$ and in the classification scheme.

MSC:
 39A12 Discrete version of topics in analysis 39A06 Linear equations (difference equations) 39A22 Growth, boundedness, comparison of solutions (difference equations) 34B24 Sturm-Liouville theory
Full Text:
References:
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