Gerald Teschl
Oscillation Theory and Renormalized Oscillation Theory for Jacobi
Operators
(64K, LaTeX2e)
ABSTRACT. We provide a comprehensive treatment of oscillation theory for Jacobi operators
with separated boundary conditions. Our main results are as follows: If
$u$ solves the Jacobi equation $(H u)(n) = a(n) u(n+1) + a(n-1) u(n-1) - b(n)
u(n) = \lambda u(n)$, $\lambda\in \mathbb R$ (in the weak sense) on an arbitrary
interval and satisfies the boundary condition on the left or right, then the
dimension of the spectral projection $P_{(-\infty, \lambda)}(H)$ of $H$ equals the
number of nodes (i.e., sign flips if $a(n)<0$) of $u$. Moreover, we present a
reformulation of oscillation theory in terms of Wronskians of solutions, thereby
extending the range of applicability for this theory; if
$\lambda_{1,2}\in \mathbb R$ and if $u_{1,2}$ solve the Jacobi equation $H
u_j= \lambda_j u_j$, $j=1,2$ and respectively satisfy the boundary condition
on the left/right, then the dimension of the spectral projection
$P_{(\lambda_1, \lambda_2)}(H)$ equals the number of nodes of the Wronskian
of $u_1$ and $u_2$. Furthermore, these results are applied to establish the
finiteness of the number of eigenvalues in essential spectral gaps of
perturbed periodic Jacobi operators.