  The class of three-diagonal Jacobi matrix with exponentially increasing
elements is considered. Under some assumptions the matrix corresponds to
unbounded self-adjoint operator in the weighted space. The weight depends on
elements of the matrix and in some cases can arise indefinite metric.
  We proved that eigenvalue problem for this operator is equivalent to the
eigenvalue problem of Sturm--Liouville operator with discrete self-similar
weight. The asymptotic formulas for eigenvalues are obtained. These formulas
differ for cases of definite and indefinite metrics.

Functional Analysis;Spectral Theory