Simple Sturm-Liouville system resembling Associated-Legendre equation?

[omyojj]

I'm trying to solve the following Sturm-Liouville system
$$\frac{d}{dx}\left((1-x^2)^2\frac{d}{dx}y\right) + (\lambda - k^2)y=0$$

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
$$y^{\prime}(a) = y^{\prime}(-a) = 0$$
plus the parity condition
$$y(x) = y(-x)$$.
(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

 

[omyojj]

One thing I tried is to integrate the above equation from x=0 to x=a to get
$$\lambda_n \int_0^a y_n dx = k^2 \int_0^a y_n dx$$
(The first term on the left-hand side vanished from the given boundary conditions.
Hence,
$$\lambda_n = k^2$$
which is strange because all the eigenvalues are given as λ_n = k^2.
Where have I been wrong?

 

[MathematicalPhysicist]

I don't think you got it wrong.

It's $$\lambda_k$$

 

[omyojj]

No.
Maybe I should explain the background to this problem.
I encountered the above equation while solving the p-mode(acoustic wave) dispersion relation in an horizontally infinite isothermal disk with vertical stratification in the z direction.
Boundary conditions are chosen so that vertical displacement at the disk boundaries become zero.

Vertical density distribution given by
$$\rho(z) = \rm{sech}^2(z)$$
or
$$\rho(x) = (1-x^2)$$
when we make use of a Lagrangian variable z = tanh(x)

y here is perturbation variable
$$y = \rho_1(x)/\rho(x)$$

λ_n is the square of frequency ω_n for horizontal Fourier wavenumber k>0.

Physically, for each horizontal wavenumber k, there would be corresponding infinite number of p-modes, each with increasing frequency ω_n and eigenfunction y_n having n zeros between z=-a and z=a.
I want to find the solution to the fundamental mode (ω_0^2 = λ_0).

Anyway, k should be regarded as a given number (like ν(nu), the index representing the order of Bessel's equation)).

 

[nuknew]

If you have not already solved your problem, use the Frobenius method with y(x) as an infinite series polynomial in x. This method is used in Math World's internet info for solving the Legendre differential equation. Also there are many other references available on the net or literature. Best wishes.

 

[JJacquelin]

This particular Sturm-Liouville equation can be solved in terms of associeted Legendre functions (see attachment)