Monotonicity of the Zeros of a Cross-Product of Bessel Functions

Our principal result is that for fixed β (0 < β 1), and fixed α > 0, the positive zeros of the cross-product \[Jν + β (x)K_ν (α x) - α ^β J_ν (x)Kν + β (α x)\] increase with ν, - β / 2 ν < ∞. In particular this implies that the eigenvalues of the boundary value problem \[∇ _n^2 p + λ ^2 g(x)p = 0,\]p radial, $p'(0) = 0$, p(∞ ) < ∞, increase with the dimension n where ∇ ₙ² is the n-dimensional Laplacian, x is the distance from the origin and $g(x) = 1$, 0 x 1, g(x) = - α ², $x > 1$.