Trouble with Laplace transform of a function

[imtitanium]

I have a probability distribution as follows:
\begin{equation}p_j(t)=\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{1,k}e^{ij\tilde{k}}\right|^2+\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{2,k}(t)e^{ij\tilde{k}}\right|^2\end{equation}
where,
\begin{equation}c_{1,k}(t)=\cos\theta_kt-\frac{i\cos\tilde{k}\sin\theta_kt}{\sqrt{1+cos^2\tilde{k}}},
\end{equation}

\begin{equation}
c_{2,k}(t)=-\frac{ie^{i\tilde{k}}\sin\theta_kt}{\sqrt{1+cos^2\tilde{k}}}.
\end{equation}

\begin{equation}
\tilde{k}=\frac{2\pi k}{N}.
\end{equation}

\begin{equation}\sin\theta_k=\frac{1}{\sqrt{2}}\sin\frac{2\pi k}{N}.\end{equation}

Can anyone please help me find the laplace transform of $p_j(t)$? I would also be glad if anyone could help me plot the probability distribution as a function of $j$ for fixed $N$ and $t$. Please note that the probability is zero for odd $j+t$. Thank You!

 

[maajdl]

I guess you need to expand all these expression until you end up with a sum of products like

sin(Θk t) sin(Θk' t)

The Laplace transform of this term is simple, but the sum still need to be simplified.
I expect the result to be expressed as sum, but a closer form is not impossible.

I this expression comes out of a piece of physics, it would be helpful to explain it here.
The underlying physics could help to figure out the end result.

 

[imtitanium]

Ok, this expression describes the probability distribution of a quantum random walker in a finite graph. And one more thing is that even Matlab cannot evaluate the laplace transform of this function. I do not need the detailed derivation of the result (although knowing it would be pretty cool), I just need the final result so that I can use it in another problem. Any help is highly appreciated.

 

[maajdl]

Should we read "sin(Θk t)" or "sin(Θk) t" ?

 

[imtitanium]

\begin{equation}\sin(\theta_kt)\end{equation}

 

[maajdl]

This Laplace transform should be useful:

$$L(Sin(a t) Sin(b t)) = \frac{2 a b p}{a^4+2 a^2 \left(p^2-b^2\right)+\left(b^2+p^2\right)^2}$$

 

[imtitanium]

Will MATLAB be able to handle this? I know we can find out laplace transform in matlab. I want to try out that, but when I type the above expression in matlab, I always get some error.