I am aware that similar questions have been answered earlier. But still I am not able to convince myself on following question:
@anna_v put it best that "weak eigenstate" is meaningless.
Popular usage favors "weak eigenbasis" or "flavor basis". The "eigen"-bit is an appeal to intuitive physics lore: we normally switch bases corresponding to diagonal forms of operators that don't commute, and we then transform to their respective eigenbases. You might have taken the "you-know, you-know" terminology a step too literally.
The unitary PMNS mixing matrix $$ \begin{bmatrix} {\nu_e} \\ {\nu_\mu} \\ {\nu_\tau} \end{bmatrix} = \begin{bmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \end{bmatrix} \begin{bmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \end{bmatrix} $$ rotates a free Hamiltonian (so, mass operator) neutrino eigenstate to the flavor basis, a fictitious "convenience" basis. The mass eigenstates are the ones that travel in nature, from supernovas to us, or from Batavia, IL, to Lead, SD, or across Japan, etc... The flavor basis states are pre-packaged combinations coupling to the charged leptons (mass eigenstates!) in your detector, so, in a way, a "laboratory artifact". In practice, you never operate on the e-neutrino combination with anything to get it back. That's why it's not nature's favorite. (It is analogous to the quark state d', the convenience combination of d, s and b quarks weakly-coupling to the up quark.)
If you must contrive a bogus operator that $\nu_e$ is an eigenstate of, with eigenvalue 1, sure, you could consider the evident projector$$|\nu_e\rangle\langle \nu_e| = |\nu_i\rangle U_{ei} U_{je}^* \langle \nu_j| ~, $$ where summation over the latin indices (1,2,3) is implied. In either basis, you see it acts on $\nu_e$ leaving it identical, but annihilates $\nu_\mu$ and $\nu_\tau$.
In the mass eigenstate basis, this follows from the unitarity of the PMNS matrix: With α representing flavor, $$ |\nu_i\rangle U_{ei} U_{je}^* \langle \nu_j| ~ |\nu_\alpha\rangle= |\nu_i\rangle U_{ei} U_{je}^* U_{j\alpha} = U_{ei}|\nu_i\rangle ~~ \delta_{e\alpha}= \delta_{e\alpha} ~|\nu_e\rangle ~. $$
But I hasten to add, nature will never bother with this projector... If one saw it in a paper, one would shake one's head at the pointlessness of the exercise. The actual charged weak coupling has no use for it, simply using the meaningful vertex $\frac{e}{\sqrt{2} \sin\theta_w} W^-_\mu \overline{\psi_\alpha}\gamma^\mu P_L U_{\alpha i}\nu_i$, etc, where $\psi_\alpha$ is the e,μ,τ triplet in whose favor this is all done. Let's stick with "flavor basis".
which operator is connected to weak eigenstates
AFAIK interactions are not connected to operators in a one to one relation. There are four interactions, the electromagnetic, the weak, the strong and the gravitational, characterized by the corresponding ( em, weak, strong) interaction coupling constant
These interaction refer to the elementary particles in the standard model of particle physics
All particles carry quantum numbers which may be conserved or not depending on the interaction.
and what is its eigenvalue.
so the term "weak eigenstate" does not have a meaning , imo.
For neutrinos there are the number operators , operating on the specific neutrino field, the creation and annihilation operators . The quantum field theory formalism has to be used and the search brings up a number of papers on how oscillations can be modeled in quantum field theory:
Flavor oscillation of traveling neutrinos is treated by solving the one-dimensional Dirac equation for massive fermions. The solutions are given in terms of squeezed coherent state as mutual eigenfunctions of parity operator and the corresponding Hamiltonian, both represented in bosonic creation and annihilation operators. It was shown that a mono-energetic state is non-normalizable, and a normalizable Gaussian wave packet, when of pure parity, cannot propagate. A physical state for a traveling neutrino beam would be represented as a normalizable Gaussian wave packet of equally-weighted mixing of two parities, which has the largest energy-dependent velocity. Based on this wave-packet representation, flavor oscillation of traveling neutrinos can be treated in a strict sense. These results allow the accurate interpretation of experimental data for neutrino oscillation, which is critical in judging whether neutrino oscillation violates CP symmetry.
It ain't simple.
