Inverse spectral theory of finite Jacobi matrices
Peter Craig Gibson · Transactions of the American Mathematical Society · 2002
We solve the following physically motivated problem: to determine all finite Jacobi matrices J J and corresponding indices i , j i,j such that the Green’s function \[ ⟨ e j , ( z I − J ) − 1 e i ⟩ \langle e_j,(zI-J)^{-1}e_i\rangle \] is proportional to an arbitrary prescribed function f ( z ) f(z) . Our approach is via probability distributions and orthogonal polynomials. We introduce what we call the auxiliary polynomial of a solution in order to factor the map \[ ( J , i , j ) ⟼ [ ⟨ e j , ( z I − J ) − 1 e i ⟩ ] (J,i,j)\longmapsto [\langle e_j,(zI-J)^{-1}e_i\rangle ] \] (where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.