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Oscillation Theory and Renormalized Oscillation Theory for Jacobi Operators

✍ Scribed by Gerald Teschl

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
1009 KB
Volume
129
Category
Article
ISSN
0022-0396

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✦ Synopsis

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 (Hu

(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 (& , *) (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 * 1, 2 # R and if u 1, 2 solve the Jacobi equation Hu j =* j u j , j=1, 2 and respectively satisfy the boundary condition on the leftÂright, then the dimension of the spectral projection P (*1, *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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