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$H = -\sum\limits_{i,j} A_{ij} c_i^{\dagger} c_j + \frac{U}{2} \sum\limits_i(c_i^\dagger c_i)(c_i^\dagger c_i -1)$ is defined to be a Hamiltonian for modeling quantum random walk of identical particles on a graph (Hubbard Model). A particle can make transition from one vertex to another if there is an edge between then and a double-occupancy charge $U$ is imposed. $A$ is the adjacency matrix of the finite graph.

I vaguely understand that first term is about transition from vertex j to i ( destroyed at j and created at i ) and second term will result in $U$ when acted on state where two particle occupy same state. How do I make this understanding rigorous and obtain the matrix elements of $H$ when there are two bosonic particles in some vertices initially. What corresponds to energy, and eigenfunction of the $H$?

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