Inverse spectral theory of finite Jacobi matrices

Gibson, Peter

doi:10.1090/S0002-9947-02-03078-7

Inverse spectral theory of finite Jacobi matrices
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by Peter C. Gibson PDF

Trans. Amer. Math. Soc. 354 (2002), 4703-4749 Request permission

Abstract:

We solve the following physically motivated problem: to determine all finite Jacobi matrices $J$ and corresponding indices $i,j$ such that the Green’s function \[ \langle e_j,(zI-J)^{-1}e_i\rangle \] is proportional to an arbitrary prescribed function $f(z)$. 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 \[ (J,i,j)\longmapsto [\langle e_j,(zI-J)^{-1}e_i\rangle ] \] (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.

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