# Hamiltonian and Green functions.

1. Apr 5, 2007

### tpm

Let be the 1-D Hamiltonian:

$$\hat H = -\hat D ^{2} + V(\hat x)$$ (1)

and its associated 'Green function' so:

$$-D^{2} G(x,s)+V(x)G(x,s)=\delta (x-s)$$ (2)

then my question is if there is a relationship between:

$$Tr[exp(-a\hat H ) ]$$ and $$Det[1-aG]$$ (3)

where a >0 and 'G' is the Kernel or Green function (due to the relationship between ODE's and Integral equation..

In case (3) is not exact i would like to know if there is any relationship between the trace of a Hamiltonian and the Determinant of its Fredholm Operator (in this case the Green function or propagator)