I Norm of a Functional and wavefunction analysis

SeM

Hi, I am working on a home-task to analyse the properties of a ODE and its solution in a Hilbert space, and in this context I have:

1. Generated a matrix form of the ODE, and analysed its phase-portrait, eigenvalues and eigenvectors, the limits of the solution and the condition number of the matrix.
2. I have however not applied the Functional analysis of the general solution, as I am not sure how to get by this.

It appears from Kreyszig "Intro to Functional Analysis" that a FUNCTIONAL can be represented in a Hilbert space. Does this mean that a FUNCTION (i.e a wavefunction) can also equally be represented in a Hilbert space?

I have calculated the inner product of the ODE matrix, and defined its neither positive or negative definite value. However, which steps should I take in order to Represent the Function and general solution of the ODE in a Hilbert space?

Thanks!
 
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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 

Stephen Tashi

Science Advisor
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It appears from Kreyszig "Intro to Functional Analysis" that a FUNCTIONAL can be represented in a Hilbert space. Does this mean that a FUNCTION (i.e a wavefunction) can also equally be represented in a Hilbert space?

If you are asking if a statement that we can do such-and-such with an arbitrary "Functional" implies that we can also do such-and-such with an arbitrary "Function" the answer is: No. A "functional" is a particular type of function. So a functional has properties that an arbitrary function need not have.

However, in the particular case you are asking about (i.e. a wave function) the answer is probably yes. For example, a function represented in a fourier series can be considered to be a vector in a Hilbert Space by regarding each of the sine and cosine functions as a basis vector in the Hilbert Space.

What does Kreyszig mean by "represented"? The coefficients of the sine and cosine functions can be regarded as "representing" the function as a vector in the Hilbert Space.
 
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SeM

If you are asking if a statement that we can do such-and-such with an arbitrary "Functional" implies that we can also do such-and-such with an arbitrary "Function" the answer is: No. A "functional" is a particular type of function. So a functional has properties that an arbitrary function need not have.

However, in the particular case you are asking about (i.e. a wave function) the answer is probably yes. For example, a function represented in a fourier series can be considered to be a vector in a Hilbert Space by regarding each of the sine and cosine functions as a basis vector in the Hilbert Space.

What does Kreyszig mean by "represented"? The coefficients of the sine and cosine functions can be regarded as "representing" the function as a vector in the Hilbert Space.

Thanks for this Stephen, I think that solves it. I will look into the Fourier transform, and then further on how that can be applied in a Hilbert space.
 

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