So, I was examining the ground state of a Bose-Hubbard dimer in the negligible interaction limit, which essentially amounts to constructing and diagonalizing a two-site hopping matrix that has the form(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

H_{i,i+1}^{(n)} = H_{i+1,i}^{(n)} = - \sqrt{i}\sqrt{n-i+1},

[/tex]

with all other elements zero. The superscript [itex]n[/itex] refers to the fixed number of particles present on the dimer, and the dimension of the matrix is given by [itex]n+1[/itex].

Essentially this gives rise to a hollow centrosymmetric tridiagonal matrix. Explicitly, we have:

[tex]

H^{(2)} =

\left(

\begin{array}{ccc}

0 & -\sqrt{2} & 0 \\

-\sqrt{2} & 0 & -\sqrt{2} \\

0 & -\sqrt{2} & 0 \\

\end{array}

\right)

[/tex][tex]

H^{(3)} =

\left(

\begin{array}{cccc}

0 & -\sqrt{3} & 0 & 0 \\

-\sqrt{3} & 0 & -2 & 0 \\

0 & -2 & 0 & -\sqrt{3} \\

0 & 0 & -\sqrt{3} & 0 \\

\end{array}

\right)

[/tex][tex]

H^{(4)} =

\left(

\begin{array}{ccccc}

0 & -2 & 0 & 0 & 0 \\

-2 & 0 & -\sqrt{6} & 0 & 0 \\

0 & -\sqrt{6} & 0 & -\sqrt{6} & 0 \\

0 & 0 & -\sqrt{6} & 0 & -2 \\

0 & 0 & 0 & -2 & 0 \\

\end{array}

\right)

[/tex] and so on.

In examining the unnormalized eigenstate with the lowest (most negative) eigenvalue, there seems to exist a Pascal-triangle-like sequence:

[tex]|\psi_{g}^{(1)}\rangle= [1,1][/tex][tex]|\psi_{g}^{(2)}\rangle= [1,\sqrt{2},1][/tex][tex]|\psi_{g}^{(2)}\rangle= [1,\sqrt{3},\sqrt{3},1][/tex][tex]|\psi_{g}^{(3)}\rangle= [1,\sqrt{4},\sqrt{6},\sqrt{4},1][/tex][tex]|\psi_{g}^{(4)}\rangle= [1,\sqrt{5},\sqrt{10},\sqrt{10},\sqrt{5},1][/tex]

This is highly suggestive that some sort of recurrence relation or mapping to binomial expansion exists; however thus far I have not been successful in trying to extract it. Might some one be able to shed some light on this?

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# Lowest eigenstate of hopping matrix

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