Completeness of the system of root functions of \(q\)-Sturm-Liouville operators. (English) Zbl 1296.39005
The authors consider the \(q\)-Sturm-Liouville operator
\[
l(y):=-\frac{1}{q}D_{q^{-1}}D_qy(x)+w(x)y(x),\quad 0\leq x\leq a<+\infty.
\]
In the above operator, it holds that \(w\) is defined on \([0,a]\) and is continuous at \(0\). Moreover, \(0<q<1\). The authors then consider \(L_0\), which is the closure of the minimal operator generated by \(l\) and then argue that the maximal operator, \(L\), is given by \(L=L_0^*\). The main result of the paper is that all eigenvalues of the operator \(L\) lie on the open upper half-plane and are purely discrete. Furthermore, a completeness result regarding the eigenfunctions of \(L\) in the space \(L_q^2(0,a)\) is established; note that \(L_q^2(0,a)\) is the space of all complex-valued functions \(f\) defined on \([0,a]\) such that
\[
\| f\|:=\left(\int_0^a |f(x)|\;d_qx\right)^{\frac{1}{2}}<+\infty.
\]
Reviewer: Christopher Goodrich (Omaha)
MSC:
39A13 | Difference equations, scaling (\(q\)-differences) |
34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |