This is a many-body hamiltonian, since the indexed operators act on each "substate" of the entire system. Imagine a total state A represented by a ket vector; then this is actually composed of $N$ states each in its corresponding Hilbert space. The total Hilbert space which is the inner product of each subspace $h_1\otimes h_2\otimes...h_n$ is totally described by A. So to sum it up, this kind of Hamiltonian will always act on states similar to A. 

To find the expectation values of H, first you need to know on what basis are you working on. For example, in problems such as nuclear structure, where rotations and concept of angular moments are important, the Wigner $D_2$ base is relevant, and this you can construct a state A totally known, which will help you solve the Schrodinger equation. 

To be more specific, look on papers of A.A. Raduta, Tanabe, Yamamoto. They always solve the eigenvalue problem for such many body hamiltonians. 

I hope this will help you!