Article
J. Differential Equations 129, 532-558 (1996)
[DOI: 10.1006/jdeq.1996.0126]
Oscillation Theory and Renormalized Oscillation Theory for Jacobi Operators
Gerald Teschl
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) = λ u(n),
λ∈ 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(-∞, λ)(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
λ1,2∈ ℝ and if u1,2 solve the Jacobi equation
H uj= λj uj, j=1,2 and respectively satisfy the boundary condition
on the left/right, then the dimension of the spectral projection
P(λ1, λ2)(H) equals the number of nodes of the Wronskian
of u1 and u2. Furthermore, these results are applied to establish the
finiteness of the number of eigenvalues in essential spectral gaps of
perturbed periodic Jacobi operators.
MSC91: Primary 39A10, 39A70; Secondary 34B24, 34L05
Keywords: Discrete oscillation theory, Jacobi operators, spectral theory
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