Sturm—Liouville problems with several parameters

We consider the regular linear Sturm-Liouville problem (second-order linear ordinary differential equation with boundary conditions at two points x = 0 and x = 1, those conditions being separated and homogeneous) with several real parameters λ₁,…,λ_N. Solutions to this problem correspond to eigenvaluesλ = (λ₁,…,λ_N) forming sets $\text{R}$^N determined by the number of zeroes in (0, 1) of solutions. We describe properties of these sets including: boundedness, and when unbounded, asymptotic directions. Using these properties some results are given for the system of N Sturm-Liouville problems which share only the parameters λ. Sharp results are given for the system of two problems sharing two parameters. The eigensurfaces for a single problem are closely related to the cone $\text{K={λ R}{}^{\mathrm{N}}\text{:λ}{}_1\text{a}{}_1\text{(x)+…+λ}{}_{\mathrm{N}}\text{a}{}_{\mathrm{N}}\text{(x)?0 for all x in 0,1]}}$, particularly in questions of boundedness. The cone K and related objects are discussed, and a result is given which relates cones with two oscillation conditions known as “Right-Definiteness” and “Left-Definiteness.”