Singular left-definite Sturm--Liouville problems (Q1886297)

This paper deals with the eigenvalues of singular left-definite Sturm-Liouville problems for the equation \[ My:=- (py')'+qy= \lambda wy\quad\text{on}\quad J=(a,b),\qquad -\infty\leq a<b \leq\infty,\tag{1} \] where \[ (i)\quad 1/p,\,q,\,w\in L^{1}_{\text{ loc}}(J;\mathbb{R}), \qquad(ii)\quad p,\,|w|>0\text{ a.e. on }J,\qquad (iii)\quad w\text{ changes sign on }J. \] Since the weight function \(w\) changes sign, the standard Sturm-Liouville theory cannot be applied to this equation. The authors eliminate this difficulty by the following observation. For \(\lambda\in \mathbb{R}\), let \(S(\lambda)\) be a selfadjoint realization of the differential expression \[ \frac{1}{|w|} (M-\lambda w) \] in the Hilbert space \(H=L^{2}(J,|w|)\) equipped with the inner product \((f,g)=\int_Jf \bar{g}|w|\). By \(D\) we denote the domain of \(S(\lambda)\). Then \(\lambda\) is in the spectrum of \((1)\) on \(D\) if and only if \(0\in\sigma (S(\lambda))\). Among other results the authors obtain the following assertion. Assume that the problem \((1)\) on \((D)\) is left-definite, i.e.,there exists \(c>0\) such that \(\int_J(Mf)\bar{f}\geq c\) for all \(f\) in \(D\cap U\), where \(U\) is the unit ball of \(H\). For \(n\in\mathbb{N}_{0}=\{0,1,2,\ldots\}\), we put \[ \xi_{n}(\lambda)=\inf_{F\in G_{n+1}(D)}\sup_{f\in F\cap U}(S(\lambda)f,f), \] where \(G_{n+1}(D)\) is the set of \(n\)-dimensional subspaces of \(D\). For each \(n\in\mathbb{N}_{0}\), the equation \(\xi_{n}(\lambda)=0\) always has exactly one positive root \(\lambda=\lambda_{n}\) and one negative root \(\lambda=\lambda_{-n}\), and they satisfy that \(|\lambda_{\pm n}|\geq\xi_{n}(0)\). (i) Suppose that for some \(n\in\mathbb{N}_{0}\), \(0\) is an eigenvalue of \(S(\lambda_{n})\). Then \(\lambda_{j}\), \(j=0,\ldots,n\), are eigenvalues of problem \((1)\) on \(D\) and satisfy \[ 0<\lambda_{0}\leq\lambda_{1}\leq\cdots\leq\lambda_{n}.\tag{2} \] Moreover, these eigenvalues are the only points of the spectrum of \((1)\) on \(D\) in \([0,\lambda_{n}]\). (ii) Suppose that for some \(m\in\mathbb{N}_{0}\), \(0\) is an eigenvalue of \(S(\lambda_{-m})\). Then \(\lambda_{-j}\), \(j=0,\ldots,m\), are eigenvalues of problem \((1)\) on \(D\) and satisfy \[ \lambda_{-m}\leq\cdots\leq\lambda_{-1}\leq\lambda_{-0}<0.\tag{3} \] Moreover, these eigenvalues are the only points of the spectrum of \((1)\) on \(D\) in \([\lambda_{-m},0]\). (iii) Suppose that for some \(m,\,n\in{\mathbf N}_{0}\), \(0\) is an eigenvalue of both \(S(\lambda_{-m})\) and \(S(\lambda_{n})\). Then \(\{\lambda_{j}|\quad j=-m,\ldots,-1,-0,0,1,\ldots,n\}\) are eigenvalues of problem \((1)\) on \(D\) and satisfy \[ \lambda_{-m}\leq\cdots\leq\lambda_{-1}\leq\lambda_{-0}<0<\lambda_{0}\leq\lambda_{1}\leq \ldots\leq\lambda_{n}.\tag{4} \] Moreover, these eigenvalues are the only points of the spectrum of \((1)\) on \(D\) in \([\lambda_{-m},\lambda_{n}]\). In \((2)\)--\((4)\) only geometrically double eigenvalues appear twice. Strict inequalities hold throughout \((2)\)--\((4)\) whenever the boundary condition is separated or nonreal coupled, or at least one endpoint is a limit point.