  We explore the sparsity of Weyl-Titchmarsh $m$-functions of discrete
Schr\"odinger operators. Due to this, the set of their $m$-functions cannot be
dense on the set of those for Jacobi operators. All this reveals why an inverse
spectral theory for discrete Schr\"odinger operators via their spectral
measures should be difficult. To obtain the result, de Branges theory of
canonical systems is applied to work on them, instead of Weyl-Titchmarsh
$m$-functions.
