Solving an equation for a unitary matrix

[Threepwood]

Homework Statement

I have an equation for a unitary matrix $$U$$,
$$\sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} - \tilde{\epsilon}_k \delta_{qm}} \right)} = 0$$
I need to solve this equation for $$U$$

Homework Equations

The property of unitarity requires that $$U\bar{U} = \hat{I}$$

The Attempt at a Solution

If $$q \neq m$$ then
$$\sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} } \right)} = 0$$
so that
$$\sum_k \left(\varepsilon_k - \mu\right)\bar{U}_{qk} U_{km} = - \gamma \sum_{kp} \bar{U}_{qk} U_{pm}$$

If $$q = m$$ then
$$\sum_k \left(\varepsilon_k - \mu\right) = - \sum_{kp} \left(U_{mk} U_{pm} - \tilde{\epsilon_k}\right)$$

How do I combine these two results in one equation for $$U$$?

 

[Jhero]

your best approach to solving this problem would be to use the given equation and the properties of unitarity to manipulate and simplify the equation. Here are some steps you can take to solve for U:

1. Use the property of unitarity (U\bar{U} = \hat{I}) to rewrite the equation as \sum_k{\left(\left(\varepsilon_k - \mu\right)U_{qk}U_{km} + \gamma \sum_p{U_{qk}U_{pm}} - \tilde{\epsilon}_k \delta_{qm} \right)} = 0.

2. Expand the sums and rearrange terms to get \sum_k{\left(\left(\varepsilon_k - \mu\right)U_{qk}U_{km}\right)} + \sum_p{\gamma U_{qk}U_{pm}} - \sum_k{\tilde{\epsilon}_k \delta_{qm}} = 0.

3. Use the fact that q \neq m to simplify the first term on the left-hand side to \sum_k{\left(\varepsilon_k - \mu\right)U_{qk}U_{km}} = -\gamma \sum_{kp}{U_{qk}U_{pm}}.

4. Substitute this into the equation to get \sum_p{\gamma U_{qk}U_{pm}} - \sum_k{\tilde{\epsilon}_k \delta_{qm}} = 0.

5. Use the fact that q = m to simplify the first term on the left-hand side to \sum_p{\gamma U_{qk}U_{pm}} = -\tilde{\epsilon}_q.

6. Substitute this into the equation to get -\tilde{\epsilon}_q - \sum_k{\tilde{\epsilon}_k \delta_{qm}} = 0.

7. Rearrange the terms to get \tilde{\epsilon}_q = -\sum_k{\tilde{\epsilon}_k \delta_{qm}}.

8. Use this result to solve for U_{qm} in terms of \tilde{\epsilon}_k: U_{qm} = \frac{-\tilde{\epsilon}_q}{\tilde{\epsilon}_m}.

9. Use this result to solve for U in terms of \tilde{\epsilon}_k: U