Evolution of neutrinos flavor states What do we mean by saying that neutrino flavor states do not satisfy the schrodinger equation? How does the time evolution of states look like?
 A: Because of the way neutrinos are (I'm going to try to keep this simple) the flavor eigenstates are related to the mass eigenstates via:
$$
\vec{\nu}_f = \bf{U} \vec{\nu}_m,
$$
where the vector $\vec{\nu_f}$ has flavor eigenstate components ($\nu_{\mu}$, $\nu_{e}$, $\nu_{\tau}$), the vector $\vec{\nu}_m$ has mass eigenstate components ($\nu_{1}$, $\nu_{2}$, $\nu_{3}$), and $\bf{U}$ is a unitary transformation matrix.  This expression is telling us that flavor eigenstates can be expressed as a linear combination of mass eigenstates.  These mass eigenstates are the ones that satisfy the schrodinger equation:
$$
\hat{H}\vec{\nu_m}=\vec{E}\vec{\nu_m},
$$
where the vector $\vec{E}$ has entries of the energy eigenvalues associated with $\vec{\nu_m}$.  
Now if we have a neutrino that begins in a flavor eigenstate ($|{\nu_e}\rangle$ for example) then we can apply the time evolution operator, $\hat{U}(t)$,to find how the flavor eigenstate evolves in time, i.e.
$$
|{\nu(t)}\rangle = \hat{U}(t)|{\nu_e}
\rangle \;.
$$
For simplicity let this be a free theory (no potential).  Now because $\hat{U}(t) = e^{iHt}$ we will expand $|{\nu_e}\rangle$ into the linear superposition of mass eigenstates and find that:
$$
|{\nu(t)}\rangle = Ae^{iE_1t}|\nu_1
\rangle+Be^{iE_2t}|\nu_2
\rangle+Ce^{iE_3t}|\nu_3\rangle \;,
$$
where the (1,2,3) subscript denotes the mass eigenstates and the constants $(A,B,C)$ are given by the unitary transformation matrix $\bf{U}$ entries.  This is clearly not an eigenstate of the Hamiltonian! Let me know if you want me to expand upon any of these ideas
A: Neutrino's flavor states evolve with time by a process called oscillation in which the three definite mass eigenstates that each flavor has associated with it exist in superposition. 
A neutrino may be created in one flavor, an electron neutrino for example, and be considered in a super position of electron, muon, and tau neutrinos with only non-zero eigenvalues for the electron neutrino component. However, as it travels (a process which takes time, letting it evolve) the eigenvalues of the muon and tau components become non-zero, and the mass becomes a mixture of the definite associated masses.   
This discovery is what the 2015 Nobel prize in physics was awarded for 
