Double Check Normalization Condition

[thatboi]

Consider the state $\ket{\Psi} = \sum_{1 \leq n_{1} \leq n_{2} \leq N} a(n_{1},n_{2})\ket{n_{1},n_{2}}$ and suppose $$|a(n_{1},n_{2})| \propto \cosh[(x-1/2)N\ln N]$$ where $0<x=(n_{1}-n_{2})/N<1$. The claim is that all $a(n_{1},n_{2})$ with $n_{2}-n_{1} > 1$ go to $0$ as $N\rightarrow\infty$. Clearly we need some kind of normalization constant, otherwise the cosh function should just blow up. So is the right normalization condition then $$C^{2}\frac{1}{4}\sum_{n_{1},n_{2}}^{N} |a(n_{1},n_{2})|^2 = 1$$ where $C$ is our normalization constant (I introduced the $1/4$ because I removed the ordering in the sum)? Because I tried doing the calculation and making the plot but I still cannot see this exponential decay.

 

[thatboi]

Ok I took another crack at the problem and this is indeed the correct normalization condition.