The characteristic function for Jacobi matrices with applications
Abstract
We introduce a class of Jacobi operators with discrete spectra which is characterized by a simple convergence condition. With any operator J from this class we associate a characteristic function as an analytic function on a suitable domain, and show that its zero set actually coincides with the set of eigenvalues of J in that domain. Further we derive sufficient conditions under which the spectrum of J is approximated by spectra of truncated finitedimensional Jacobi matrices. As an application we construct several examples of Jacobi matrices for which the characteristic function can be expressed in terms of special functions. In more detail we study the example where the diagonal sequence of J is linear while the neighboring parallels to the diagonal are constant.
 Publication:

arXiv eprints
 Pub Date:
 January 2012
 arXiv:
 arXiv:1201.1743
 Bibcode:
 2012arXiv1201.1743S
 Keywords:

 Mathematics  Spectral Theory;
 47B36;
 15A18;
 47A11
 EPrint:
 1 figure