A criterion for spectrum decomposition (Q1810238)

The author considers the Sturm-Liouville problem \[ -k^{-2}y''+ q(x) y=\lambda y,\quad x\in (0,1),\quad y(0)= y(1)= 0. \] Here, \(k\) is a positive parameter, and \(q\) is a real-valued function twice continuously differentiable on \([0,1]\). The author obtains a necessary and sufficient condition for the decomposition of the spectrum of an arbitrary nonsymmetric potential \(q(x)\) whose least value is attained at finitely many points.