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

The $m$-functions of discrete Schr\"odinger operators are sparse compared to those for Jacobi operators