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'

 

[tacman]

Pseudo differential operators are a type of operator commonly used in mathematical analysis and partial differential equations. They are defined as a linear combination of differential operators with smooth coefficients. These operators are particularly useful when dealing with infinite-dimensional ODEs, as they allow for the manipulation of functions that may not have a well-defined derivative.

In the given equation, the operator f(\partial_x) involves an infinite-dimensional ODE, meaning that it contains an infinite number of derivatives. This can make it difficult to solve for a general solution. However, by making the ansatz y(x)=e^{ax}, we are able to simplify the equation and find a solution in terms of the coefficients of the operator.

When h(x)=0, the resulting equation becomes a homogeneous one, and the ansatz y(x)=e^{ax} satisfies it. This means that for non-zero h(x), we can construct an orthonormal basis using the solutions given by the ansatz to express the solution on the interval (0,c).

As for the expression G(x,y), it is not clear what the operator in the denominator represents without further context. However, in general, we can give a meaning to the expression by interpreting it as a kernel or a generalized function. In this case, the integration over the volume element dV suggests that G(x,y) is a type of distribution, which can be useful in solving certain types of problems in mathematical physics.