Can Zero Be Considered an Eigenfunction?

[pivoxa15]

Homework Statement

If y(x)=0 satisfies the ode and all the boundary conditions than does it count as the first eigenfunction?

The Attempt at a Solution

It wouldn't satisfy the orthogonality relation though? In that the integral of 0 and 0 is 0 even though the integral is over two eigenfunctions that are the same.

 

[Dick]

A linear operator acting on the zero vector is always the zero vector. This doesn't count as 'eigen' behavior. Otherwise every linear operator would have zero eigenvalue. Read a carefully stated definition of eigenfunction.

 

[pivoxa15]

Good point. I should always relate to linear algebra. The 0 vector is never an eigenvector as it would imply the determinant of the matrix is nonzero.

So the 0 function is never the eigenfunction.