How Do Delta Functions Affect Eigenvalues and Eigenfunctions in ODEs?

[Combinatorics]

Homework Statement

Find negative eigenvalues and corresponding eigenfunctions to the following operator:
$H:= - \frac{d^2}{dx^2} - \delta_{-r} -2\delta{r}$ .
(The eigenfunction should be twice contiously differentiable, except for possible jump discontinuities at $+-r$ of the first and second derivatives. In addition, the eigenfunctions $f$, must satisfy that $f, f' , f'' = O(1/|x|)$

Homework Equations

The Attempt at a Solution

I really have no idea about it... I found this question on the web-
http://www.harding.edu/lmurray/Quantum_files/_Ch5%20Delta%20Function%20Potential.pdf
(problem 5.1)
and I can't figure out how to generalize the calculation for$r=0$ to this situation...

Your help is needed!


Thanks !

 

[mighty2000]

Dear fellow scientist,

Thank you for bringing this problem to our attention. After reviewing the forum post and the problem, I believe I have a solution that may be helpful to you.

First, let's start by defining the operator H as:

H := -\frac{d^2}{dx^2} - \delta_{-r} - 2\delta_r

where \delta_{-r} and \delta_r represent the Dirac delta function at points -r and r, respectively.

To find the eigenvalues and eigenfunctions of this operator, we can use the general method for solving the Schrödinger equation with a delta function potential. This involves finding the solutions to the following equation:

H\psi(x) = \lambda\psi(x)

where \psi(x) is the eigenfunction and \lambda is the eigenvalue.

Next, we can substitute the definition of H into the Schrödinger equation and rearrange to obtain:

-\frac{d^2}{dx^2}\psi(x) - \delta_{-r}\psi(x) - 2\delta_r\psi(x) = \lambda\psi(x)

We can then solve this differential equation using the method of Green's functions. This involves finding the Green's function for the operator H, which is given by:

G(x,x') = \begin{cases} -\frac{1}{2}\left(e^{|x-x'|} - e^{-(x+x')}\right) & \text{if } |x| \geq r \\ -\frac{1}{2}\left(e^{-(x-x')} - e^{(x+x')}\right) & \text{if } |x| < r \end{cases}

Using this Green's function, we can then write the general solution for \psi(x) as:

\psi(x) = c_1G(x,x') + c_2G(x,x') + c_3

where c_1, c_2, and c_3 are constants to be determined.

To find the eigenvalues and eigenfunctions, we must impose the appropriate boundary conditions. In this case, we want the eigenfunctions to be twice continuously differentiable and satisfy the condition that \psi, \psi', and \psi'' are all O(1/|x|). This means that at the points -r and r, the eigenfunctions must have jump