- #1

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- Homework Statement
- This question is about the differential operator acting on functions of x in the range x ∈ [0, ∞). This is a generalization of the case covered in the notes where the range of x is finite. Here, one end of the range of the variable x is infinite.

How can I know these coefficients a_k? and get the corresponding eigenfunctions?

- Relevant Equations
- Consider the inhomogeneous eigenfunction equation: L tilde y = lambda y

we may define an operator in self-adjoint form L = wL tilde by means of a suitable weight function w(x) and the eigenfunction equation above becomes: Ly = lambda wy

We assume that we have boundary conditions on our functions that make L self-adjoint.

I have found that w(x) should be e^-x to make L self-adjoint.

and insert back get xL''+(x+1)L' +lambda L = 0

now it needs to assume a monic polynomial function, so I assume Ln = x^n+ sum from k=0 to n-1 (a_k*x^k)

get the 1st and 2nd order differential and insert back

I get lambda_n = (-nx^(n-1)*(n+x)-sum from k=0 to n-1 (a_k*k*x^(k-1)*(k+x))/(x^n + sum from k=0 to n-1 (a_k*x^k))

for n=0,1,2, i get lambda = 1, -(x+1)/(x+a_0), (-2x(x+2)-a_1(x+1))/(x^2+a_0+a_1x)

**how can i know these coefficients a_k? and how can i get the corresponding eigenfunctions?**