I have been running into a question for which I found no reference in the litterature. I do not have a strong background in number theory ; for me this is motivated by a question in PDEs (how close can be two eigenfrequencies of the disk). As such I would appreciate detailed answers and references.
Say $J_\mu$ is the $\mu$-th Bessel function, where $\mu$ is an (nonnegative) integer (one could also consider half-integer), and $j_{\mu,k}$ is its $k$-th positive zero. It has been proved by Siegel that $j_{\mu,k}$ is a transcendental number and this is enough to prove that for any distinct integer $\mu,\nu$, and any $k,l\in\mathbb{N}$, $j_{\mu,k}\neq j_{\nu,l}$. The method (given for instance in Watson's "Treatise on the Theory of Bessel functions" paragraph 15.28) is that if $\nu>\mu$ then there is a relation $$J_{\nu}(z)=J_\mu(z)R_{\nu-\mu,\mu}(z)-J_{\mu-1}(z)R_{\nu-\mu-1,\mu+1}(z)$$ where $(R_{m,\mu})$ is an explicit sequence of polynomials with rationnal coefficients, so a common zero to both $J_{\nu}$ and $J_\mu$ is also common to $J_{\mu-1}$, which can easily be shown to be impossible by looking at the ODE verified by $J_{\mu -1}$.
My question is the following: is there a quantified version of this result ? I am thinking of something of the sort $$|j_{\mu,k}-j_{\nu,l}|\geq c j_{\mu,k}^{-\alpha}$$ for some universal constants $c,\alpha>0$.
To me this seems linked to how far from algebraic number are the zeroes of Bessel function: taking again the method above it seems sufficient to have a bound from below of $R_{\nu-\mu-1,\mu+1}(j_{\nu,l})$ to get that when two zeroes of $J_\mu,J_{\nu}$ are close, then $J_{\mu}$ and $J_{\mu-1}$ are both small, which seems more manageable.
Is there any work related to this ? I am aware that the proof of transcendance of $j_{\mu,k}$ is similar to the one of $\pi$, and that there are bounds for the "irrationality measure" of $\pi$, however this would necessitate a result not only stronger for each zero (how close it is to algebraic numbers, instead of rationals) that is also uniform. Any reference or suggestion would be appreciated !
