Eigencurves for two-parameter self-adjoint ordinary differential equations of even order

AbstractThe eigenvalue problem Ly(x) = (λƒ(x) − μr(x))y(x) is considered for a self-adjoint 2nth -order operator L which is bounded below with compact resolvent on L2(a, b), and for real functions f, r with certain restrictions. It is shown that the (λ, γ) eigencurves are piecewise analytic with no line segments unless s = f/r is constant a.e., when the curves are lines, and possesses left (right) hand asymptotes iff s attains its minimum (maximum) value a.e. on some interval. The asymptote equations are given explicitly in terms of the eigenvalues of certain simpler boundary value problems. The second-order case is discussed in detail