Neutrino Oscillation in matter

[ChrisVer]

Hi, I have some problem in deriving $\Delta m_M^2$ as given in eq.35 here:
http://www.slac.stanford.edu/econf/C040802/papers/L004.PDF

When I tried to derive the eigenvalues of $H_M$ (eq.33) I got:

$m^2 = (\cos 2 \theta -x)^2 + \sin^2 2 \theta$
which is only one eigenvalue. Any help? In particular it seems he defines:
$\Delta m_M^2 = \Delta m_i^2 \times |m|$

 

[ChrisVer]

Here's my solution

 

[ChrisVer]

If I say:
$m_{1M} = m_1 + m \Rightarrow m_{1M}^2 = m_1^2 + m^2 + 2 m_1 m$
$m_{2M} = m_2 -m \Rightarrow m_{2M}^2 = m_2^2 + m^2 - 2 m_2 m$
Then:
$\Delta m_M^2 =m_{1M}^2 - m_{2M}^2 = (m_1^2 - m_2^2 ) + 2 m (m_1 +m_2 ) = \Delta m^2 + 2 m (\Sigma m)$