- #1
omyojj
- 37
- 0
I'm trying to solve the following Sturm-Liouville system
[tex]
\frac{d}{dx}\left((1-x^2)^2\frac{d}{dx}y\right) + (\lambda - k^2)y=0
[/tex]
defined in an interval -a<x<a (or 0<x<a) with 0<a<=1.
Here, k is a real number and λ is the eigenvalue of the system.
y satisfies boundary conditions
[tex]y^{\prime}(a) = y^{\prime}(-a) = 0[/tex]
plus the parity condition
[tex]y(x) = y(-x)[/tex].
(or y'(a) = 0 and y'(0) = 0)
Can anybody give me any hint on how to obtain ground state(Lower-bound eigenvalue and the corresponding eigenfunction) solution, say y_0 and λ_0?
Of course being able to obtain general solution would be much better.
Thanks
[tex]
\frac{d}{dx}\left((1-x^2)^2\frac{d}{dx}y\right) + (\lambda - k^2)y=0
[/tex]
defined in an interval -a<x<a (or 0<x<a) with 0<a<=1.
Here, k is a real number and λ is the eigenvalue of the system.
y satisfies boundary conditions
[tex]y^{\prime}(a) = y^{\prime}(-a) = 0[/tex]
plus the parity condition
[tex]y(x) = y(-x)[/tex].
(or y'(a) = 0 and y'(0) = 0)
Can anybody give me any hint on how to obtain ground state(Lower-bound eigenvalue and the corresponding eigenfunction) solution, say y_0 and λ_0?
Of course being able to obtain general solution would be much better.
Thanks