How Do Pseudo Differential Operators Solve Infinite-Dimensional ODEs?

[Klaus_Hoffmann]

let be the operator involving an infinite-dimensional ODE

$$f( \partial _{x}) y(x)=h(x)$$

then if h(x)=0 i make the ansatz $$y(x)=e^{ax}$$ so

$$\sum_{\rho } e^{x\rho}$$ $$f(\rho) =0$$

for h(x) different from '0' we construct an orthonormal basis with the solutions given above to give an expression on the interval (0,c)

Another question,.. can we give a 'meaning' to the expression.

$$G(x,y)= \int dV \frac{e^{ik|x-y|}{E-Ak^{2}-V(\partial _{k})}$$

 

[Klaus_Hoffmann]

i meant the operator

$$\int_{a}^{b}dx \frac{exp(-iux)}{x+f(\partial)-5}$$

where the derivative is respect to 'x'