Solving Boundary Value Problems: Are Eigenvalues Equal?

[zetafunction]

let be the two boundary value problem

$$-D^{2}y(x)+f(x)y(x)= \lambda _{n} y(x)$$

with $$y(0)=0=y(\infty)$$

and the same problem $$-D^{2}y(x)+f(x)y(x)= \beta _{n} y(x)$$

with $$y(-\infty)=0=y(\infty)$$

i assume that in both cases the problem is SOLVABLE , so my question is , are the eigenvalues in both cases equal ? , i mean $$\lambda _{n} = \beta _{n}$$ , or have the same dependence on parameter 'n' ?

 

[sshzp4]

I don't understand your notation. You first homogenize your ODE, then you find the eigenvalues of your characteristic equation. Then you apply BCs to extract constants of integration. So you find eigenvalues before applying BCs. So, in general, yes.

However, the way you have described your equations, your characteristic equation will be different, so different eigenvalues. So if I don't understand your query, then my answer is probably wrong.