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*of_{(-∞, λ)}(H)*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*λ*and if_{1,2}∈ ℝ*u*solve the Jacobi equation_{1,2}*H u*,_{j}= λ_{j}u_{j}*j=1,2*and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection*P*equals the number of nodes of the Wronskian of_{(λ1, λ2)}(H)*u*and_{1}*u*. Furthermore, these results are applied to establish the finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Jacobi operators._{2}
** MSC91:** Primary 39A10, 39A70; Secondary 34B24, 34L05

**Keywords:** *Discrete oscillation theory, Jacobi operators, spectral theory*

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