96-36 Gerald Teschl

Oscillation Theory and Renormalized Oscillation Theory for Jacobi Operators (64K, LaTeX2e) Feb 5, 96

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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.

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