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

Our principal result is that for fixed $\beta (0 < \beta \leqq 1)$, and fixed $\alpha > 0$, the positive zeros of the cross-product \[J_{\nu + \beta } (x)K_\nu (\alpha x) - \alpha ^\beta J_\nu (x)K_{\nu + \beta } (\alpha x)\] increase with $\nu $, ${{ - \beta } / {2 \leqq \nu < \infty }}$. In particular this implies that the eigenvalues of the boundary value problem \[\nabla _n^2 p + \lambda ^2 g(x)p = 0,\]p radial, $p'(0) = 0$, $p(\infty ) < \infty $, increase with the dimension n where $\nabla _n^2 $ is the n-dimensional Laplacian, x is the distance from the origin and $g(x) = 1$, $0 \leqq x \leqq 1$, $g(x) = - \alpha ^2 $, $x > 1$.