  Lieb, Schultz and Mattis (LSM) studied the S=1/2 XXZ spin chain. Theorems of
LSM's paper can be applied to broader models. In the original LSM theorem it
was assumed the nonfrustrating system. However, reconsidering the LSM theorem,
we can extend the LSM theorem for frustrating systems.
  Next, several researchers have tried to extend the LSM theorem for excited
states. In the cases $S^{z}_{T} = \pm 1,\pm 2 \cdots$, the lowest energy
eigenvalues are continuous for wave number $q$. But we find that their proofs
are insufficient, and we improve them.
  In addition, we can prove the LSM theory without the assumption of the
discrete symmetry, which means that the LSM type theorems are applicable for
Dzyaloshinskii-Moriya type interactions or other nonsymmetric models.
