BESSEL FUNCTIONS IN A QUANTUM-BILLIARD CONFIGURATION PROBLEM
Keyword(s):
In studying various quantum-billiard configurations, R. L. Liboff (J. Math. Phys.35 (1994) 2218), was led to investigate the vanishing of f(ν)=j2ν,1 - jν2, where jμk is the kth positive zero of the Bessel function Jμ(x). Here we show that the even more general function fα(ν)=cαν,k - cν,k+l is increasing and vanishes once (and only once) in 0<ν<∞, provided α≥π/2 and [Formula: see text], k, l=1,2,3,…. As usual, cμn is the nth positive zero of the cylinder function Cμ(x)=Jμ (x) cos θ - Yμ(x) sin θ. Specialized to Liboff's case, f(ν), this yields not only the existence of a zero of f(ν) but also its uniqueness.
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