Explicit sharp bounds for all nodes of Sturm-Liouville operators with potentials in $L^1$ balls

For the classical Sturm-Liouville operators, we prove the sharp bounds for all nodes of eigenfunctions by regarding these nodes as nonlinear functionals of potential $q\in L^1[0,1]$. By studying the optimization problems to minimize or to maximize the nodes $\{T_{i,m}\}$ subject to the constraint $\mathrm{\parallel }q{\mathrm{\parallel }}_1=r$ with $r>0$ and using the strong continuity of the nodes in potentials, we obtain the explicit expressions for the sharp bounds, which are given as elementary functions.