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

On spectrum of Jacobi operator with exponentially increasing matrix elements