Does this eigenvalue problem has degenerate solutions?

[Urkel]

Hello everyone,
I am solving an eigenvalue problem. Right now, I would like to know;
How to determine the degeneracy of eigensolution of sturm-liouville differential eigenvalue problem?
I have an eigenvalue sturm-liouville problem H f(y) = E f(y) where H is a differential operator and E is the eigenvalue that can be re-written as the following high order PDE

[(E + a ∂_y^2 - b y) (E - a ∂_y^2- b y) + k^2 ]f(y)=0

a, b are constants. ∂_y^2 is 2nd derivative w.r.t. y
How to determine if the eigenvalue is degenerate with respect to k, that is, it does not depend on k? k is constant.
That is, can we find several eigenfunctions that correspond to the same eigenvalue E?

How to prove that the eigenvalue is degenerate or not? Any theorem on that?

I hope someone who is expert in theory of PDE can help me on this.

Thanks

 

[jvicens]

for your question! Determining the degeneracy of an eigensolution in a Sturm-Liouville differential eigenvalue problem is an important aspect of solving these types of problems. There are several methods and theorems that can help us determine if an eigenvalue is degenerate or not.

One approach is to use the Sturm-Liouville theorem, which states that for a regular Sturm-Liouville problem, the eigenvalues are simple and distinct. This means that each eigenvalue has a unique corresponding eigenfunction. Therefore, if we can show that the eigenvalues in this problem are not simple and distinct, then we can conclude that they are degenerate.

Another approach is to use the Wronskian, which is a mathematical tool used to determine linear independence of functions. If we can show that the Wronskian of the eigenfunctions corresponding to a particular eigenvalue is equal to zero, then we can conclude that the eigenvalue is degenerate.

Additionally, there are certain theorems, such as the Sturm-Liouville degeneracy theorem, that provide conditions for the degeneracy of eigenvalues in a Sturm-Liouville problem. These theorems can be applied to your problem to determine if the eigenvalue is degenerate or not.

In summary, there are several methods and theorems that can help us determine the degeneracy of an eigenvalue in a Sturm-Liouville differential eigenvalue problem. It is important to carefully analyze the problem and choose the appropriate method to prove the degeneracy or non-degeneracy of the eigenvalue. I hope this helps in your research!