Pseudo differential operators

In summary, the conversation discusses the use of the operator f(\partial_x) in solving an infinite-dimensional ODE, with the ansatz y(x)=e^{ax} being used for h(x)=0. For h(x) not equal to 0, an orthonormal basis is constructed using the solutions from the ansatz to express the solution on the interval (0,c). The conversation then moves on to discussing the expression G(x,y) involving an integral and the meaning behind it.
  • #1
Klaus_Hoffmann
86
1
let be the operator involving an infinite-dimensional ODE

[tex] f( \partial _{x}) y(x)=h(x) [/tex]

then if h(x)=0 i make the ansatz [tex] y(x)=e^{ax} [/tex] so

[tex] \sum_{\rho } e^{x\rho} [/tex] [tex] f(\rho) =0 [/tex]

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.

[tex] G(x,y)= \int dV \frac{e^{ik|x-y|}{E-Ak^{2}-V(\partial _{k})} [/tex]
 
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  • #2
i meant the operator

[tex] \int_{a}^{b}dx \frac{exp(-iux)}{x+f(\partial)-5} [/tex]

where the derivative is respect to 'x'
 
  • #3


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.
 

1. What are pseudo differential operators?

Pseudo differential operators are a type of mathematical operator that combines elements of both differential and integral operators. They are used in the study of partial differential equations and have many applications in physics, engineering, and other fields.

2. How do pseudo differential operators differ from regular differential operators?

Pseudo differential operators differ from regular differential operators in that they involve non-local operations and can be defined on spaces of functions with more general growth conditions. They also have a wider range of symbols, including symbols that are not smooth functions.

3. What are the applications of pseudo differential operators?

Pseudo differential operators have many applications in areas such as signal processing, quantum mechanics, and fluid dynamics. They are also used in the study of elliptic and hyperbolic partial differential equations, and in the analysis of boundary value problems.

4. How are pseudo differential operators related to Fourier transforms?

Pseudo differential operators and Fourier transforms are closely related, as the Fourier transform of a pseudo differential operator with a symbol of order m is a pseudo differential operator with a symbol of order -m. This relationship allows for the application of Fourier transforms in the study of pseudo differential operators.

5. What are the properties of pseudo differential operators?

Pseudo differential operators have many useful properties, including linearity, associativity, and commutativity. They also have a calculus of symbols, which allows for the composition and inversion of operators. Additionally, pseudo differential operators have a well-defined action on Sobolev spaces, which makes them useful in the study of partial differential equations.

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