Spectral Deformations of Jacobi Operators

We extend recent work concerning isospectral deformations for one-dimensional Schrödinger operators to the case of Jacobi operators. We provide a complete spectral characterization of a new method that constructs isospectral deformations of a given Jacobi operator (Hu)(n) = a(n)u(n + 1) + a(n − 1)u(n − 1) − b(n)u(n). Our technique is connected to Dirichlet data, that is, the spectrum of the operator H∞ n0 on ` 2(−∞, n0) ⊕ `(n0,∞) with a Dirichlet boundary condition at n0. The transformation moves a single eigenvalue of H∞ n0 and perhaps flips which side of n0 the eigenvalue lives. On the remainder of the spectrum the transformation is realized by a unitary operator.