Help with Eigenfunctions: Find Eigenvalue-Eigenfunction Pair

In summary, eigenfunctions are special functions that remain unchanged when operated on by a linear operator, while eigenvalues are the corresponding scalars associated with the eigenfunctions. Finding eigenvalue-eigenfunction pairs is important in various areas of mathematics and science, as they provide insight into the behavior and properties of systems. To find these pairs, one must solve an eigenvalue problem, which can be done through methods like matrix diagonalization or using the characteristic equation. It is possible for a system to have multiple eigenvalue-eigenfunction pairs, as different eigenfunctions can have the same eigenvalue. These pairs have practical applications in quantum mechanics, signal processing, and solving boundary value problems in differential equations.
  • #1
chappyform
1
0
eigenfunctions. help!

I would be very grateful for any help on the following question:

Find any single eigenvalue-eigenfunction pair, with a real eigenvalue, for the
following operator:

[itex]\textit{L}[/itex] = ([itex]\partial^2[/itex]/[itex]\partial[/itex]x) + ([itex]\partial[/itex]/[itex]\partial[/itex]x) + 2Id

subject to the initial boundary conditions u(0)=u(pi)=0
 
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  • #2
welcome to pf!

hi chappyform! welcome to pf! :wink:

show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 
  • #3


Since the derivatives are only with respect to x, you don't really need the partial derivatives.
 
  • #4


what is 2Id?
 
  • #5
X89codered89X said:
what is 2Id?

twice the identity! :wink:
 

FAQ: Help with Eigenfunctions: Find Eigenvalue-Eigenfunction Pair

What are eigenfunctions and eigenvalues?

Eigenfunctions are special functions that remain unchanged when operated on by a linear operator. Eigenvalues are the corresponding scalars that are associated with the eigenfunctions.

Why do we need to find eigenvalue-eigenfunction pairs?

Finding eigenvalue-eigenfunction pairs is important in many areas of mathematics and science, including quantum mechanics, differential equations, and signal processing. These pairs provide insight into the behavior and properties of systems.

How do you find eigenvalue-eigenfunction pairs?

To find eigenvalue-eigenfunction pairs, you need to solve an eigenvalue problem, which involves finding the eigenvalues and corresponding eigenfunctions that satisfy a specific equation. This can be done through various methods, such as matrix diagonalization or using the characteristic equation.

Can there be multiple eigenvalue-eigenfunction pairs for a given system?

Yes, a system can have multiple eigenvalue-eigenfunction pairs. This is because different eigenfunctions can have the same eigenvalue, and vice versa.

How can eigenvalue-eigenfunction pairs be used in practical applications?

Eigenvalue-eigenfunction pairs have many practical applications, such as in quantum mechanics, where they are used to describe the energy states of particles. They are also used in signal processing to analyze and manipulate signals, and in differential equations to solve boundary value problems.

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