Oscillation theory and renormalized oscillation theory for Jacobi operators
For the difference operator \[ (Hu)(n)= a(n)u(n+1)- a(n-1)u(n-1)- b(n)u(n) \] the dimension of the spectral projection \(P_{(-\infty,\lambda)}(H)\) is expressed in terms of nodes of any solution of the equation \(Hu=\lambda u\). For example: if \(\lambda\) is an eigenvalue of \(H\), then \[ \text{dim Ran }P_{(-\infty,\lambda)} (H)=\#(u_+(\lambda)), \] where \(\#(u_+(\lambda))\) denotes the total numbers of nodes of the nontrivial solution \(u\) satisfying a boundary condition at \(\infty\). Moreover (reformulating the oscillation theory) the dimension of \(P_{(\lambda_1,\lambda_2)} (H)\) is considered. It is shown that this dimension is equal to the number of nodes of the Wronskian \(W(u_1,u_2)\) of two solutions \(u_1,u_2\) satisfying the equations \(Hu_j= \lambda_ju_j\), \(j=1,2\). As application of the main results the spectra of perturbed periodic Jacobi operators are studied.
scientific article; zbMATH DE number 933284