A generalization of an inverse problem on a second order boundary value problem
The instability intervals of the differential equation \((1)\quad y''+[\lambda a(t)-q(t)]y=0\) are considered where a and q are \(\pi\)- periodic and integrable over (0,\(\pi\)). An instability interval is an open interval such that for each \(\lambda\) in it at least one solution of the differential equation is unstable. The endpoints of these intervals can be characterized as eigenvalues of the two associated boundary value problems \(y(0)=y(\pi)\), \(y'(0)=y'(\pi)\) and \(y(0)=-y(\pi)\), \(y'(0)=- y'(\pi).\) There is always an unbounded interval of instability. But there might be no bounded or finitely many bounded intervals of instability. The author derives necessary conditions such that all but n finite instability intervals of (1) vanish. This result generalizes the case of Hill's equation where \(a=1\), in which case the conditions are also sufficient.
scientific article; zbMATH DE number 4214399