Discrete spectra criteria for singular difference operators. (English) Zbl 0936.39008

The author investigates oscillatory and spectral properties of the difference operator \[ B(y)={(-1)^n\over w_k}\Delta ^n\left (p_k\Delta ^n y_k\right), \tag{*} \] where \(w_k\) is a positive weight function. The results of the paper are based on a Wirtinger-type inequality applied to a certain discrete quadratic functional associated with the operator \(B\). Explicit conditions on the sequences \(p,w\) are given which guarantee that the equation \(B(y)=y\) is nonoscillatory. This result is then used to study sufficient conditions for discreteness and boundedness below (the so-called property BD) of self-adjoint operators generated by \(B\).
Reviewer: O.Došlý (Brno)


39A70 Difference operators
47A10 Spectrum, resolvent
47B39 Linear difference operators
39A11 Stability of difference equations (MSC2000)
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