Hello, here the Omega=5/2 is the only 'true' quantum numbers, the others in square parentheses are not good quantum number, they are indicated as 'asymptotic' quantum numbers, i.e. they are strictly valid only in some limits. They are used to recall, from where the states come from. You can find more details in books such as K.Heyde "Basic ideas and concepts in nuclear physics" or Ring-Schuck.

Anyway during the non-crossing, the energies never touch, but the character of the states (its composition in terms of amplitudes of the spherical basis states) does indeed exchange (see ch. 11 of cited Heyde book).

you ask me, I ask who?

It would be more natural to ask (and answer) why there some energies which do cross. The answer would then be that there is some underlying symmetry which prevents mixing of  two approaching states.

The generic case is eigenvalue repulsion, which prevents states from crossing. Since that is just a name, consider a simple example. Assume

H0(t) =  1 + t sigmaz, where sigmaz is the diagonal Pauli matrix.

Since this is already diagonal, it is obvious the the eigenvalues, 1+t, 1-t, will cross when t passes zero. A symmetry which allows this is invariance under rotation around the z-axis.

Next add a small perturbative term, c sigmax, with c very, very small (but non-zero). This breaks the above rotation symmetry. The eigenvalues become [1+sqrt(c2+t2)], [1-sqrt(c 2+t2)] . Increasing t from negative values, this means that they will approach each other as before until t becomes roughly -5c, then turn around until t becomes roughly 5c, after which they move away from each other like before. It is also interesting to analyse the behaviour of the eigenvectors, and to solve the time-dependent equation

-i epsilon d/dt |ps(t)> = H(t) |psi(t)>

on some interval symmetric around t=0, for various values of epsilon. To observe how an initial eigenvector may "jump the gap" when epsilon is large (sudden approximation), but follow the  instantaneous t-dependent eigenvector when epsilon is small (adiabatic limit).

One can understand that this is a general feature, by considering the quantum mechanical formula for second order perturbation theory, and note how it behaves when another unperturbed eigenvalue becomes very close to the perturbed one. Then the denominator becomes very small, so the perturbation inevitably becomes very large -- unless the matrix elements in the numerator is zero.

An interesting numerical experiment is to calculate the eigenvalues of a Hermitian matrix with independent, random elements, and plot the position of the eigenvalues, viewed as particles placed along a line. The statistical distribution of such configurations behaves  as described by statistical mechanics of particles with a repulsive potential.