Discrete spectra criteria for certain classes of singular differential and difference operators
Criteria are established for discreteness and boundedness from below of the spectrum (the so-called property BD) for singular differential and difference operators of the form \[ m(y)\equiv {(-1)^n\over w(t)}\bigl(r(t) y^{(n)} \bigr)^{(n)}\quad \text{and} \quad b(y)_{k+n} \equiv{(-1)^n\over w_k} \Delta^n (r_k\Delta^n y_k), \] where \(n\in\mathbb{N}\), \(k\in\dot J=[n_0,\infty)\), \(n_0 \in \mathbb{N}\), the sequences \(r_k^{-1}\) and \(w_k\) are positive, real, and summable on \(\dot J\) while \(t\in \dot I=[a,+\infty)\), \(a\in\mathbb{R}\), and \(r^{-1}\), \(w\in L_{\text{loc}}(\dot I)\) are positive real functions.
singular differential and difference operators
scientific article; zbMATH DE number 1755267
scientific article