Asymptotic behavior of sums of the type $\sum_{k=m}^{n-1}\exp(i\omega\sqrt{nk})$ as $n,m\to\infty$

The asymptotic behavior as $n,m\to\infty$ of the sum
$$ \sum_{k,l=m}^{n-1}\exp[i\omega\sqrt{n}(\sqrt{k}+\sqrt{l})]\Phi\biggl(1-\frac{|\sqrt{k}-\sqrt{l}|}{\Delta}\biggr), $$
is studied where $\Phi(t)=0$ for $t\leqslant0$ and $t$ for $t>0$.