Exact results on spin dynamics and multiple quantum NMR dynamics in alternating spin-$1/2$ chains with $XY$-Hamiltonian at high temperatures

We extend the picture of a transfer of nuclear spin-$1/2$ polarization along a homogeneous one-dimensional chain with the $XY$-Hamiltonian to the inhomogeneous chain with alternating nearest neighbour couplings and alternating Larmor frequencies. To this end, we calculate exactly the spectrum of the spin-$1/2$ $XY$-Hamiltonian of the alternating chain with an odd number of sites. The exact spectrum of the $XY$-Hamiltonian is also applied to study the multiple quantum (MQ) NMR dynamics of the alternating spin-$1/2$ chain. MQ NMR spectra are shown to have the MQ coherences of zero and $\pm$ second orders just as in the case of a homogeneous chain. The intensities of the MQ coherences are calculated.