Undergrad 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!

 

[Stephen Tashi]

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.

 

[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.