N site Hamiltonians

Question

Hello, I have a question I need to solve in my physics and I have absolutely no Idea how to solve it.

Any chance someone can help me with the solution?

Given a system of three sites arranged in a chain. The standard basis is | x⟩ with x = −1,0,1. In addition, the system is symmetrical for mirroring. Such a system can be described (after calibration) using only two parameters.

1) Write down the matrix that represents a mirroring operation.
2) Continuing from the previous section define a base of self-states for mirroring | 0⟩, | +⟩, | −⟩

3) If Hamiltonian is registered in the standard base - what will be the free parameters?
4) Write down what the Hamiltonian looks like at the base defined in Section 2.
5) Suppose you add an option to jump between the first and last site. In addition suppose there is no magnetic field. How it will affect the Hamiltonian you have listed in sections 3 and 4.

Tip: The result of section 4 can be clarified on the basis of symmetry considerations. The system has a "trivial" self-state whose identity is not affected by the values ​​of the parameters. Who is this situation? The system makes a "trivial" reduction to the problem of only two modes.