Heat kernel (PDE) asymptotic expansion

[zetafunction]

let be the PDE eigenvalue problem $$\partial_{t} f =Hf$$

then if we define its Heat Kernel $$Z(u)= \sum_{n=0}^{\infty}e^{-uE_{n}}$$ valid only for positive 'u'

then my question is how could get an asymptotic expansion of the Heat Kernel as u approaches to 0

$$Z(u) \sim \sum_{n=0}^{\infty}a_{n} u^{n}$$ valid as u-->0+ (zero by the right)

 

[matematikawan]

I did asked similar question in the past. It's quite a paradox!

https://www.physicsforums.com/showthread.php?t=308414