(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have an equation for a unitary matrix [tex]U[/tex],

[tex]\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[/tex]

I need to solve this equation for [tex]U[/tex]

2. Relevant equations

The property of unitarity requires that [tex]U\bar{U} = \hat{I}[/tex]

3. The attempt at a solution

If [tex]q \neq m[/tex] then

[tex]\sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} } \right)} = 0[/tex]

so that

[tex]\sum_k \left(\varepsilon_k - \mu\right)\bar{U}_{qk} U_{km} = - \gamma \sum_{kp} \bar{U}_{qk} U_{pm}[/tex]

If [tex]q = m[/tex] then

[tex]\sum_k \left(\varepsilon_k - \mu\right) = - \sum_{kp} \left(U_{mk} U_{pm} - \tilde{\epsilon_k}\right)[/tex]

How do I combine these two results in one equation for [tex]U[/tex]?

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# Solving an equation for a unitary matrix

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