On the spectra of infinite-dimensional Jacobi matrices
Let \(\ell_ 2\supset D(J)\to^{J}\ell_ 2\) be a self-adjoint operator with the representation \[ Je_ n=a(n+1)e_{n+1}+b(n)e_ n+a(n)e_{n- 1},\quad (n=1,2,\dots);\quad Je_ 0=a(1)e_ 1+b(0)e_ 0. \] By using the Green function, a general comparison equation is constructed; and this equation is invoked to obtain an upper bound on the number of eigenvalues of \(J\) using Bargmann's argument. Also, by using a modification of the Birman-Schwinger argument, an alternative upper bound on the number of eigenvalues of \(J\). It is also pointed out that this bound is valid for matrix orthogonal polynomials.
upper bound on the number of eigenvalues
scientific article; zbMATH DE number 4098088
scientific article