Floquet theory for left-definite Sturm-Liouville problems
Let \(1/p,q,w\in L_{\text{loc}}(J,\mathbb R)\), \(p,| w| >0\) a.e., where \(J=[a,\infty )\) or \(J=(-\infty ,\infty )\) and \(w\) changes sign. The operator \(T\) on \(H=L^2(J,| w| )\) is defined by \[ Tf=\frac1w[-(pf')'+qf], \] with the usual domain, where in the case \(J=[a,\infty )\) also a boundary condition \(A_1f(a)+A_2(pf')(a)=0\) is imposed. It is assumed that \(p\), \(q\), \(w\) are periodic with a fundamental period \(r\), \(0<r<\infty \). Letting \(\xi_n^P(\lambda) \) and \(\xi_n^S(\lambda) \), \(n=0,1,\dots\), denote the periodic eigenvalues of the auxiliary eigenvalue problem \[ -(py')'+(q-\lambda w)y=\xi| w| y \quad \text{on }[a,a+r], \] with \(\xi\) being the eigenvalue parameter here, it follows that \(\xi_n^P(\lambda )=0\) [\(\xi_n^S(\lambda )=0\)] has exactly one positive and one negative root, denoted by \(\lambda _{\pm n}^P\) [\(\lambda _{\pm n}^S\)]. Under the assumption \(\xi_0^P(0)>0\), it is shown that the essential spectrum of \(T\) is the union of the intervals \(I_{\pm n}\), with \(I_{\pm n}=[\lambda _{\pm n}^P,\lambda _{\pm n}^S]\) for \(+\) and \(n\) even or \(-\) and \(n\) odd, and \(I_{\pm n}=[\lambda _{\pm n}^S,\lambda_{\pm n}^P]\) otherwise.
scientific article; zbMATH DE number 2162015