Exact ground states for a class of one-dimensional frustrated quantum spin models

We have found the exact ground state for two frustrated quantum spin-$\frac{1}{2}$ models on a linear chain. The first model describes the antiferromagnet-ferromagnet transition point. The singlet state at this point has double-spiral ordering. The second model is equivalent to special case of the spin-$\frac{1}{2}$ ladder. It has a nondegenerate singlet ground state with exponentially decaying spin correlations, and there is an energy gap. The exact ground-state wave function of these models is presented in a special recurrent form, and a recurrence technique for expectation value calculations is developed.