jmc

First note that $$\lceil x\rceil -\lfloor x\rfloor =\{\begin{array}{ll}1& \text{if x is not an integer},\\ 0& \text{if x is an integer}.\end{array}$$Thus for any positive integer $k$, $$\lceil {\log }_{\sqrt{2}}k\rceil -\lfloor {\log }_{\sqrt{2}}k\rfloor =\{\begin{array}{ll}1& \text{if k not an integer power of 2},\\ 0& \text{if k an integer power of 2}.\end{array}$$The integers $k$, $1\le k\le 1000$, that are integer powers of $\sqrt{2}$ are described by $k=2^j$, $0\le j\le 9$. Thus $$\sum _{k=1}^{1000}k(\lceil {\log }_{\sqrt{2}}k\rceil -\lfloor {\log }_{\sqrt{2}}k\rfloor )=\sum _{k=1}^{1000}k-\sum _{j=0}^9{2}^j=\frac{1000\cdot 1001}{2}-1023=\boxed{499477}.$$