Gerald Teschl
Spectral Deformations of Jacobi Operators
(35K, LaTeX2e)
ABSTRACT. We extend recent work concerning isospectral deformations
for one-dimensional Schr\"odinger 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 $(H u)(n) = a(n) u(n+1) +
a(n-1)
u(n-1) - b(n) u(n)$. Our technique is connected to Dirichlet data, that is, the
spectrum of the operator $H^\infty_{n_0}$ on $\ell^2 (-\infty,n_0) \oplus
\ell^2
(n_0,\infty)$ with a Dirichlet boundary condition at $n_0$. The transformation
moves a single eigenvalue of $H^\infty_{n_0}$ and perhaps flips which side of
$n_0$ the eigenvalue lives. On the remainder of the spectrum the transformation
is realized by a unitary operator.