Help How to get green function of Bessel's differential equation?

[inflaton]

In my project, we enconter such kind of bessel's differential equation with stochastic source, like

$$\Phi''+\frac{1+2\nu}{\tau}\Phi'+k^2\Phi=\lambda\psi(\tau)$$

where we use prime to denote the derivative with $$\tau$$, $$\nu$$
and $$\lambda$$ are real constant parameter.

how to get the green function of bessel's differential equation?

 

[kosovtsov]

I do not think that you need the Green function to solve your problem.

If you can solve homogeneous linear DE, then you can easily write out the general solution to the corresponding non-homogeneous linear DE (see http://arxiv.org/abs/math-ph/0409035" ).

In your case

$$\Phi(\tau) = \tau^{-\nu}J_\nu}(k\tau)\,C1+\tau^{-\nu}Y_\nu(k\tau)\,C2-\frac{1}{2}\tau^{-\nu}\pi\lambda[\int\,-\tau^{\nu+1}J_\nu(k\tau)\psi(\tau)\,d\tau\,Y_\nu(k\tau)+\int\,\tau^{\nu+1}Y_\nu(k\tau)\psi(\tau)\,d\tau\,J_\nu(k\tau)]$$

If you nevertheless do like the Green function, substitute $$\psi(\tau)=\delta(\tau-\tau_0)$$ to the above expression.

 

[inflaton]

I do not think that you need the Green function to solve your problem.

If you can solve homogeneous linear DE, then you can easily write out the general solution to the corresponding non-homogeneous linear DE (see http://arxiv.org/abs/math-ph/0409035" ).

In your case

$$\Phi(\tau) = \tau^{-\nu}J_\nu}(k\tau)\,C1+\tau^{-\nu}Y_\nu(k\tau)\,C2-\frac{1}{2}\tau^{-\nu}\pi\lambda[\int\,-\tau^{\nu+1}J_\nu(k\tau)\psi(\tau)\,d\tau\,Y_\nu(k\tau)+\int\,\tau^{\nu+1}Y_\nu(k\tau)\psi(\tau)\,d\tau\,J_\nu(k\tau)]$$

If you nevertheless do like the Green function, substitute $$\psi(\tau)=\delta(\tau-\tau_0)$$ to the above expression.

Thanks! You are master!