Consider the operator $\mathcal{A} = -\Delta =-\frac{1}{r}\frac{d^2}{dr^2}r $ (spherical coordinates with symmetry). I am looking for eigenvalues $\lambda_k$, eigenfunctions $\varphi_k$ for it.
$$\mathcal{A}\varphi_k=\lambda_k\varphi_k \iff$$ $$\varphi''_k + \frac{2}{r}\varphi_k'+\lambda_k\varphi_k = 0.$$
Now, I'm looking at a solution of this problem and at this step the author considers this equation the "Bessel Equation" and uses the ready solutions for it and if I do this to, I get the correct answer. However, in my table of formulae the bessel equation is defined as $$u''+\frac{1}{r}u'+\left(\lambda-\frac{v^2}{r^2}\right)u = 0$$ I realize that $v$ can be $0$. But how is this definition consistent with what I have above, with the $2$ instead of a $1$ in the second term?