I Double Check Normalization Condition

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