The Cooper Problem in tight binding

[mcas]

Homework Statement

Find the coupling energy of a Cooper pair in a 1D lattice in tight binding (nearest neighbors), i.e. solve the Cooper Problem.

Relevant Equations

An equation to solve:
$1=\lambda \sum_{\vec{k}} |w_{\vec{k}}|^2 \frac{1}{E-2\xi_{\vec{k}}}$ for ${\xi_\vec{k} \geq 0}$
where:
$\lambda = const$ is a part of a seperable potential $V_{\vec{k}\vec{k'}}$
$\xi_{\vec{k}} = \varepsilon_{\vec{k}}-\mu$
$w_{\vec{k}}$ is $1$ for $\hbar w_D \geq \xi_{\vec{k}} \geq 0$ and $0$ in any other case

We have a one dimensional lattice with a lattice constant equal to a (I'm omitting vector notation because we are in 1D). The reciprocal lattice vector is k_n=n\frac{2 \pi}{a}.
So to get the nearest neighbour approximation I need to sum over k = -\frac{2 \pi}{a}, 0, \frac{2 \pi}{a}.
If I understand everything correctly this would be the equation I need to solve for E:
1=\lambda ( |w_{\frac{2 \pi}{a}}|^2 \frac{1}{E-2\xi_{\frac{2 \pi}{a}}} + |w_{0}|^2 \frac{1}{E-2\xi_{0}} + |w_{-\frac{2 \pi}{a}}|^2 \frac{1}{E-2\xi_{-\frac{2 \pi}{a}}})

I'm not sure if this is the right approach. If so, would the values of |w_{i}| equal 1 in this equation? Do we know the values of \xi_{i}?

 

[member 553083]

Yes, this is the correct approach. The values of |w_{i}| should equal 1, and the values of \xi_{i} should be given in the problem statement or can be calculated from the corresponding energies.