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

Peña, Simón

Math. Bohem. 124, No. 1, 35-44 (1999).

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)

MSC:

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

Keywords:

discrete spectrum; property BD; discrete variational principle; discrete Wirtinger’s inequality; singular difference operators; oscillation