  We study the average size of shifted convolution summation terms related to
the problem of Quantum Unique Ergodicity on ${\rm SL}_2 (\mathbbm{Z})\backslash
\mathbbm{H}$. Establishing an upper-bound sieve method for handling such sums,
we achieve an unconditional result which suggests that the average size of the
summation terms should be sufficient in application to Quantum Unique
Ergodicity. In other words, cancellations among the summation terms, although
welcomed, may not be required. Furthermore, the sieve method may be applied to
shifted sums of other multiplicative functions with similar results under
suitable conditions.
