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Zbl 1145.47303
Wilson, Robert Howard
Non-self-adjoint difference operators and their spectrum. (English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2057, 1505-1531 (2005). ISSN 1364-5021; ISSN 1471-2946/e

Summary: Initially, this paper is a discrete analogue of the work of {\it B. M. Brown}, {\it D. K. R. McCormack}, {\it W. D. Evans} and {\it M. Plum} [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1984, 1235--1257 (1999; Zbl 0944.34018)] on second-order differential equations with complex coefficients. That is, we investigate the general non-self-adjoint second-order difference expression $M_{x_n}=-\Delta (p_{n-1}\Delta x_{n-1})+q_nx_n$, $n\in N_0$ where the coefficients $p_n$ and $q_n$ are complex and $\Delta$ is the forward difference operator, i.e. $\Delta x_n=x_{n+1}-x_n$. Properties of the so-called Hellinger-Nevanlinna m-function for the recurrence relation $M_{x_n}=\lambda w_nx_n$, where the $w_n$ are real and positive, are examined, and relationships between the properties of the $m$-function and the spectrum of the associated operator are explored. However, an essential difference between the continuous and the discrete case arises in the way in which we define the operator natural to the problem. Nevertheless, analogous results regarding the spectrum of this operator are obtained.

MSC 2000:: *47B39 Difference operators (operator theory)
39A70 Difference operators
47A10 Spectrum and resolvent of linear operators

Keywords: difference operators; spectral theory; Hellinger-Nevanlinna $m$-function; limit circle; limit point

Citations: Zbl 0944.34018

Cited in: Zbl pre06122148 Zbl 1228.39006

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