Let's denote $J_\alpha$ the Bessel functions of first kind, satisfying the equation $$x^2y''+xy'+(x^2-\alpha^2)y=0$$ Now consider its zeros, there are $2$ questions.
- For the case $\alpha=0$, find the smallest $l$, such that when a zero point $x$ given, there's another zero in $(x,x+l)$
- For the case $\alpha>\frac{1}{2}$, prove that in any open interval of length $\pi$, there are at most a zero point in it.
I googled the asymptotic behavior of $J_\alpha$, which implies $l\geq\pi$. And then I can do nothing.
There are some series expansion of zero point in terms of $\alpha$. If you use this, please denote the source so that I can learn from it.
Thanks in advance.
Bessel differential equation is a particular case of a Sturm-Liouville equation.
It is true that $J_0$ has rather specific properties ; it can be placed apart with a behavior rather close to sine function.
For the general case, there is an important result called Sturm Separation Theorem which you will find explained with the particular case of Bessel differential equation treated as well here.